Precalculus Quiz: Sine And Cosine Of Complementary Angles
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Sine And Cosine Of Complementary AnglesQuestion 1 of 20

Using the cofunction identity for complementary angles, which equation correctly expresses the relationship between sine and cosine?​

sin⁡(θ)=cos⁡(90∘−θ)\sin(\theta)=\cos(90^\circ-\theta)
sin⁡(θ)=sin⁡(90∘−θ)\sin(\theta)=\sin(90^\circ-\theta)
sin⁡(θ)=cos⁡(180∘−θ)\sin(\theta)=\cos(180^\circ-\theta)
sin⁡(θ)=cos⁡(θ)\sin(\theta)=\cos(\theta)
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Precalculus Quiz

Precalculus Quiz: Sine And Cosine Of Complementary Angles

Practice Sine And Cosine Of Complementary Angles in Precalculus with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Sine And Cosine Of Complementary Angles, giving you a quick way to practice the rules, question types, and explanations that matter most for Precalculus.

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Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

Using the cofunction identity for complementary angles, which equation correctly expresses the relationship between sine and cosine?​

  1. sin⁡(θ)=cos⁡(90∘−θ)\sin(\theta)=\cos(90^\circ-\theta) (correct answer)
  2. sin⁡(θ)=sin⁡(90∘−θ)\sin(\theta)=\sin(90^\circ-\theta)
  3. sin⁡(θ)=cos⁡(180∘−θ)\sin(\theta)=\cos(180^\circ-\theta)
  4. sin⁡(θ)=cos⁡(θ)\sin(\theta)=\cos(\theta)
Explanation: This question tests understanding of the relationship between sine and cosine of complementary angles. The cofunction relationship between sine and cosine—that sin(θ) = cos(90° - θ)—derives from the fact that in a right triangle, the side that is opposite to one acute angle is adjacent to the complementary acute angle. The equation sin(θ) = cos(90° - θ) holds for all angles θ, meaning that if we know the sine of any angle, we automatically know the cosine of its complement (the angle that when added to θ gives 90°). Choice A is correct because it correctly states the cofunction relationship with sin(θ) = cos(90° - θ). Choice C confuses complementary angles (sum to 90°) with supplementary angles (sum to 180°), using 180° - θ instead of 90° - θ. Don't confuse complementary (sum to 90°) with supplementary (sum to 180°)—for supplementary angles, there is no simple cofunction relationship between sine and cosine. Key to complementary angles: remember that two angles are complementary if they sum to 90° (or π/2), and the cofunction relationship sin(θ) = cos(90° - θ) allows you to convert between sine and cosine using the complement.

Question 2

In a right triangle, the acute angles are α\alpha and β\beta, and α+β=90∘\alpha+\beta=90^\circ. Which statement must be true?

  1. sin⁡(α)=sin⁡(β)\sin(\alpha)=\sin(\beta)
  2. cos⁡(α)=cos⁡(β)\cos(\alpha)=\cos(\beta)
  3. sin⁡(α)=cos⁡(β)\sin(\alpha)=\cos(\beta) (correct answer)
  4. sin⁡(α)=cos⁡(180∘−β)\sin(\alpha)=\cos(180^\circ-\beta)
Explanation: This question tests understanding of the relationship between sine and cosine of complementary angles. In a right triangle, the two acute angles are always complementary (they sum to 90°), and by similarity, the side ratios reveal that the sine of one acute angle equals the cosine of the other acute angle. In a right triangle with acute angles α and β, where α + β = 90°, consider a side that is opposite to angle α (making sin(α) = opposite/hypotenuse). That same side is adjacent to angle β (making cos(β) = adjacent/hypotenuse = opposite/hypotenuse), showing sin(α) = cos(β). Choice C is correct because it correctly states that sin(α) = cos(β) when angles α and β are complementary in a right triangle. Choice A incorrectly claims sin(α) = sin(β), which would only be true if the angles were equal, not complementary. In right triangles, always remember that the two acute angles are complementary, so sin of one angle equals cos of the other angle—this is why cosine starts with 'co' (for complement).

Question 3

Using the complementary angle relationship, if sin⁡(30∘)=12\sin(30^\circ)=\tfrac{1}{2}, what is cos⁡(60∘)\cos(60^\circ)?

  1. 32\tfrac{\sqrt{3}}{2}
  2. 22\tfrac{\sqrt{2}}{2}
  3. 12\tfrac{1}{2} (correct answer)
  4. 11
Explanation: This question tests understanding of the relationship between sine and cosine of complementary angles. Two angles are complementary if their sum equals 90° (or π/2 radians), and there is a fundamental relationship: sin(θ) = cos(90° - θ) and cos(θ) = sin(90° - θ) for any angle θ. Since 30° and 60° are complementary (they sum to 90°), we know that sin(30°) = cos(60°). Given that sin(30°) = 1/2, we can immediately conclude that cos(60°) = 1/2. Choice C is correct because it correctly applies the complementary angle relationship to find that cos(60°) = sin(30°) = 1/2. Choice A gives √3/2, which is actually cos(30°) or sin(60°), confusing which angle pairs with which value. For special angles, use the complementary pairs: 30° and 60° (or π/6 and π/3) are complementary, so sin(30°) = cos(60°) = 1/2 and sin(60°) = cos(30°) = √3/2.

Question 4

In a right triangle, one acute angle is 45∘45^\circ, making the other acute angle also 45∘45^\circ (they are complementary). Using the complementary angle relationship, what is cos⁡(45∘)\cos(45^\circ) if sin⁡(45∘)=22\sin(45^\circ)=\tfrac{\sqrt{2}}{2}?

  1. 12\tfrac{1}{2}
  2. 32\tfrac{\sqrt{3}}{2}
  3. 22\tfrac{\sqrt{2}}{2} (correct answer)
  4. 00
Explanation: This question tests understanding of the relationship between sine and cosine of complementary angles. Two angles are complementary if their sum equals 90∘90^\circ (or π/2\pi/2 radians), and there is a fundamental relationship: sin⁡(θ)=cos⁡(90∘−θ)\sin(\theta) = \cos(90^\circ - \theta) and cos⁡(θ)=sin⁡(90∘−θ)\cos(\theta) = \sin(90^\circ - \theta) for any angle θ\theta. For the complementary angles both at 45∘45^\circ (since 45∘+45∘=90∘45^\circ + 45^\circ = 90^\circ), we apply the relationship: sin⁡(45∘)=cos⁡(45∘)\sin(45^\circ) = \cos(45^\circ) and both equal 22\tfrac{\sqrt{2}}{2}, demonstrating the cofunction property with exact values. Choice C is correct because it correctly states cos⁡(45∘)=22\cos(45^\circ) = \tfrac{\sqrt{2}}{2} using the given sin⁡(45∘)=22\sin(45^\circ) = \tfrac{\sqrt{2}}{2} and the complementary relationship. Choice B uses the wrong special triangle value, confusing the 30-60-90 ratios with the 45-45-90 ratios. For special angles, use the complementary pairs: 30∘30^\circ and 60∘60^\circ (or π/6\pi/6 and π/3\pi/3) are complementary, so sin⁡(30∘)=cos⁡(60∘)=12\sin(30^\circ) = \cos(60^\circ) = \tfrac{1}{2} and sin⁡(60∘)=cos⁡(30∘)=32\sin(60^\circ) = \cos(30^\circ) = \tfrac{\sqrt{3}}{2}, but for 45∘45^\circ, it is its own complement.

Question 5

In a right triangle, one acute angle is 25∘25^\circ and the other is 65∘65^\circ, so the angles are complementary. If cos⁡(25∘)\cos(25^\circ) is known, which expression is equal to it by the cofunction relationship?​​​

  1. sin⁡(65∘)\sin(65^\circ) (correct answer)
  2. cos⁡(65∘)\cos(65^\circ)
  3. sin⁡(25∘)\sin(25^\circ)
  4. sin⁡(155∘)\sin(155^\circ)
Explanation: This question tests understanding of the relationship between sine and cosine of complementary angles. Two angles are complementary if their sum equals 90° (or π/2 radians), and there is a fundamental relationship: sin(θ) = cos(90° - θ) and cos(θ) = sin(90° - θ) for any angle θ. Since 25° and 65° are complementary (they sum to 90°), we know that cos(25°) = sin(65°). Choice A is correct because it correctly applies cos(25°) = sin(65°) using the cofunction relationship. Choice B claims cos(25°) = cos(65°), using the same function for both angles, when the cofunction relationship requires switching from cosine to sine. In right triangles, always remember that the two acute angles are complementary, so sin of one angle equals cos of the other angle—this is why cosine starts with 'co' (for complement). Key to complementary angles: remember that two angles are complementary if they sum to 90° (or π/2), and the cofunction relationship sin(θ) = cos(90° - θ) allows you to convert between sine and cosine using the complement.

Question 6

In a right triangle ABCABC with a right angle at CC, the acute angles satisfy A+B=90∘A + B = 90^\circ. Using the complementary angle relationship, how does sin⁡(A)\sin(A) relate to cos⁡(B)\cos(B)?

  1. sin⁡(A)=cos⁡(180∘−B)\sin(A) = \cos(180^\circ - B)
  2. sin⁡(A)=sin⁡(B)\sin(A) = \sin(B)
  3. sin⁡(A)=cos⁡(B)\sin(A) = \cos(B) (correct answer)
  4. sin⁡(A)=sec⁡(B)\sin(A) = \sec(B)
Explanation: This question tests understanding of the relationship between sine and cosine of complementary angles. In a right triangle, the two acute angles are always complementary (they sum to 90°), and by similarity, the side ratios reveal that the sine of one acute angle equals the cosine of the other acute angle. In a right triangle with acute angles A and B, where A + B = 90°, consider a side that is opposite to angle A (making sin(A) = opposite/hypotenuse). That same side is adjacent to angle B (making cos(B) = adjacent/hypotenuse = opposite/hypotenuse), showing sin(A) = cos(B). Choice C is correct because it correctly applies sin(A) = cos(B) since A and B are complementary. Choice A confuses complementary angles (sum to 90°) with supplementary angles (sum to 180°), using 180° - θ instead of 90° - θ. In right triangles, always remember that the two acute angles are complementary, so sin of one angle equals cos of the other angle—this is why cosine starts with 'co' (for complement).

Question 7

In a right triangle, one acute angle measures 35∘35^\circ. The other acute angle is complementary to it (so the two acute angles sum to 90∘90^\circ). Using the cofunction identity, which equation is true?​​

  1. sin⁡(35∘)=sin⁡(55∘)\sin(35^\circ)=\sin(55^\circ)
  2. cos⁡(35∘)=cos⁡(55∘)\cos(35^\circ)=\cos(55^\circ)
  3. sin⁡(35∘)=cos⁡(55∘)\sin(35^\circ)=\cos(55^\circ) (correct answer)
  4. sin⁡(35∘)=cos⁡(145∘)\sin(35^\circ)=\cos(145^\circ)
Explanation: This question tests understanding of the relationship between sine and cosine of complementary angles. Two angles are complementary if their sum equals 90° (or π/2 radians), and there is a fundamental relationship: sin(θ) = cos(90° - θ) and cos(θ) = sin(90° - θ) for any angle θ. Since 35° and 55° are complementary (they sum to 90°), we know that sin(35°) = cos(55°). Choice C is correct because it correctly identifies that sin(35°) = cos(55°), applying the cofunction relationship where the sine of an angle equals the cosine of its complement. Choice A incorrectly claims sin(35°) = sin(55°), using the same function for both angles, when the cofunction relationship requires switching from sine to cosine. Key to complementary angles: remember that two angles are complementary if they sum to 90° (or π/2), and the cofunction relationship sin(θ) = cos(90° - θ) allows you to convert between sine and cosine using the complement. In right triangles, always remember that the two acute angles are complementary, so sin of one angle equals cos of the other angle—this is why cosine starts with 'co' (for complement).

Question 8

Using the fact that sin⁡(π6)=12\sin\left(\tfrac{\pi}{6}\right)=\tfrac{1}{2} and that π6\tfrac{\pi}{6} and π3\tfrac{\pi}{3} are complementary (sum to π2\tfrac{\pi}{2}), determine cos⁡(π3)\cos\left(\tfrac{\pi}{3}\right).

  1. 32\tfrac{\sqrt{3}}{2}
  2. 12\tfrac{1}{2} (correct answer)
  3. 00
  4. 11
Explanation: This question tests understanding of the relationship between sine and cosine of complementary angles. Two angles are complementary if their sum equals 90° (or π/2 radians), and there is a fundamental relationship: sin(θ) = cos(90° - θ) and cos(θ) = sin(90° - θ) for any angle θ. For the complementary angles π/6 and π/3, we apply the relationship: sin(π/6) = cos(π/3) and both equal 1/2, while sin(π/3) = cos(π/6) and both equal √3/2, demonstrating the cofunction property with exact values. Choice B is correct because it correctly states cos(π/3) = 1/2 using the given sin(π/6) = 1/2 and the complementary relationship. Choice A uses the wrong special triangle value, confusing the 30-60-90 ratios with the 45-45-90 ratios. For special angles, use the complementary pairs: 30° and 60° (or π/6 and π/3) are complementary, so sin(30°) = cos(60°) = 1/2 and sin(60°) = cos(30°) = √3/2. Key to complementary angles: remember that two angles are complementary if they sum to 90° (or π/2), and the cofunction relationship sin(θ) = cos(90° - θ) allows you to convert between sine and cosine using the complement.

Question 9

In a right triangle, one acute angle is 45∘45^\circ, making the other acute angle also 45∘45^\circ (they are complementary). Using the complementary angle relationship, what is cos⁡(45∘)\cos(45^\circ) if sin⁡(45∘)=22\sin(45^\circ)=\tfrac{\sqrt{2}}{2}?​

  1. 12\tfrac{1}{2}
  2. 32\tfrac{\sqrt{3}}{2}
  3. 22\tfrac{\sqrt{2}}{2} (correct answer)
  4. 00
Explanation: This question tests understanding of the relationship between sine and cosine of complementary angles. Two angles are complementary if their sum equals 90° (or π/2 radians), and there is a fundamental relationship: sin(θ) = cos(90° - θ) and cos(θ) = sin(90° - θ) for any angle θ. For the complementary angles both at 45° (since 45° + 45° = 90°), we apply the relationship: sin(45°) = cos(45°) and both equal √2/2, demonstrating the cofunction property with exact values. Choice C is correct because it correctly states cos(45°) = √2/2 using the given sin(45°) = √2/2 and the complementary relationship. Choice B uses the wrong special triangle value, confusing the 30-60-90 ratios with the 45-45-90 ratios. For special angles, use the complementary pairs: 30° and 60° (or π/6 and π/3) are complementary, so sin(30°) = cos(60°) = 1/2 and sin(60°) = cos(30°) = √3/2, but for 45°, it is its own complement.

Question 10

In the right triangle ABCABC with ∠C=90∘\angle C = 90^\circ, the acute angles AA and BB are complementary. Why does sin⁡(A)\sin(A) equal cos⁡(B)\cos(B) in this triangle?

  1. Because the side opposite AA is the same as the side adjacent to BB, so oppositehypotenuse\frac{\text{opposite}}{\text{hypotenuse}} for AA equals adjacenthypotenuse\frac{\text{adjacent}}{\text{hypotenuse}} for BB. (correct answer)
  2. Because sin⁡(A)=cos⁡(A)\sin(A)=\cos(A) for all acute angles.
  3. Because complementary angles add to 180∘180^\circ, so sine and cosine match.
  4. Because sin⁡(A)=adjacenthypotenuse\sin(A)=\frac{\text{adjacent}}{\text{hypotenuse}} and cos⁡(B)=oppositehypotenuse\cos(B)=\frac{\text{opposite}}{\text{hypotenuse}}.
Explanation: This question tests understanding of the relationship between sine and cosine of complementary angles. The cofunction relationship between sine and cosine—that sin⁡(θ)=cos⁡(90∘−θ)\sin(\theta) = \cos(90^\circ - \theta)—derives from the fact that in a right triangle, the side that is opposite to one acute angle is adjacent to the complementary acute angle. In a right triangle with acute angles A and B, where A + B = 90°, consider a side that is opposite to angle A (making sin⁡(A)=oppositehypotenuse\sin(A) = \frac{\text{opposite}}{\text{hypotenuse}}). That same side is adjacent to angle B (making cos⁡(B)=adjacenthypotenuse=oppositehypotenuse\cos(B) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{\text{opposite}}{\text{hypotenuse}}), showing sin(A) = cos(B). Choice A is correct because it correctly identifies complementary angles sum to 90∘90^\circ and explains the side relationships. Choice C confuses complementary angles (sum to 90∘90^\circ) with supplementary angles (sum to 180∘180^\circ), using 180∘−θ180^\circ - \theta instead of 90∘−θ90^\circ - \theta. In right triangles, always remember that the two acute angles are complementary, so sin⁡\sin of one angle equals cos⁡\cos of the other angle—this is why cosine starts with 'co' (for complement).

Question 11

In a right triangle, angle AA is 45∘45^\circ and angle BB is 45∘45^\circ (so AA and BB are complementary). Which equation is true?

  1. sin⁡(45∘)=cos⁡(45∘)\sin(45^\circ)=\cos(45^\circ) (correct answer)
  2. sin⁡(45∘)=cos⁡(135∘)\sin(45^\circ)=\cos(135^\circ)
  3. sin⁡(45∘)=sin⁡(90∘−45∘)\sin(45^\circ)=\sin(90^\circ-45^\circ)
  4. cos⁡(45∘)=cos⁡(90∘−45∘)\cos(45^\circ)=\cos(90^\circ-45^\circ)
Explanation: This question tests understanding of the relationship between sine and cosine of complementary angles. In a right triangle, the two acute angles are always complementary (they sum to 90°), and by similarity, the side ratios reveal that the sine of one acute angle equals the cosine of the other acute angle. Since A and B are both 45° and A + B = 90°, we know that sin(45°) = cos(45°). Choice A is correct because it correctly applies sin(A) = cos(B) for complementary angles A and B. Choice B confuses complementary angles (sum to 90°) with supplementary angles (sum to 180°), using 180° - θ instead of 90° - θ. In right triangles, always remember that the two acute angles are complementary, so sin of one angle equals cos of the other angle—this is why cosine starts with 'co' (for complement). For special angles, use the complementary pairs: 45° and 45° are complementary, so sin(45°) = cos(45°) = √2/2.

Question 12

In a right triangle, the acute angles are complementary. If one acute angle is 25∘25^\circ, which expression is equal to cos⁡(25∘)\cos(25^\circ) by the complementary angle relationship?

  1. sin⁡(65∘)\sin(65^\circ) (correct answer)
  2. sin⁡(155∘)\sin(155^\circ)
  3. cos⁡(65∘)\cos(65^\circ)
  4. sin⁡(25∘)\sin(25^\circ)
Explanation: This question tests understanding of the relationship between sine and cosine of complementary angles. In a right triangle, the two acute angles are always complementary (they sum to 90°), and by similarity, the side ratios reveal that the sine of one acute angle equals the cosine of the other acute angle. Since 25° and 65° are complementary (they sum to 90°), we know that sin(25°) = cos(65°), so cos(25°) = sin(65°). Choice A is correct because it correctly applies sin(θ) = cos(90° - θ) to show cos(25°) = sin(65°). Choice B confuses complementary angles (sum to 90°) with supplementary angles (sum to 180°), using 180° - θ instead of 90° - θ. In right triangles, always remember that the two acute angles are complementary, so sin of one angle equals cos of the other angle—this is why cosine starts with 'co' (for complement). Key to complementary angles: remember that two angles are complementary if they sum to 90°, and the cofunction relationship sin(θ) = cos(90° - θ) allows you to convert between sine and cosine using the complement.

Question 13

On the unit circle, the point at angle θ\theta is (cos⁡(θ),sin⁡(θ))(\cos(\theta),\sin(\theta)). Using the fact that 30∘30^\circ and 60∘60^\circ are complementary, what is sin⁡(60∘)\sin(60^\circ) if cos⁡(30∘)=32\cos(30^\circ)=\tfrac{\sqrt{3}}{2}?

  1. 12\tfrac{1}{2}
  2. 22\tfrac{\sqrt{2}}{2}
  3. 32\tfrac{\sqrt{3}}{2} (correct answer)
  4. 00
Explanation: This question tests understanding of the relationship between sine and cosine of complementary angles. On the unit circle, the coordinates at angle θ are (cos(θ), sin(θ)), and at the complementary angle 90° - θ, the coordinates exhibit a swap: (sin(θ), cos(θ)), demonstrating that sin(θ) = cos(90° - θ). For the complementary angles 30° and 60°, we apply the relationship: sin(30°) = cos(60°) = 1/2, while sin(60°) = cos(30°) = √3/2. Choice C is correct because it correctly applies sin(60°) = cos(30°) = √3/2. Choice B uses the wrong special triangle value, confusing the 30-60-90 ratios with the 45-45-90 ratios. For special angles, use the complementary pairs: 30° and 60° are complementary, so sin(30°) = cos(60°) = 1/2 and sin(60°) = cos(30°) = √3/2. To verify the complementary angle relationship on the unit circle, observe that as you move from angle θ to angle 90° - θ, the x and y coordinates swap positions, showing cos(θ) ↔ sin(90° - θ).

Question 14

On the unit circle, the point at angle θ\theta is (cos⁡(θ),sin⁡(θ))(\cos(\theta),\sin(\theta)). If two angles are complementary, θ\theta and 90∘−θ90^\circ-\theta, which comparison correctly matches the cofunction relationship?​

  1. cos⁡(θ)=cos⁡(90∘−θ)\cos(\theta)=\cos(90^\circ-\theta)
  2. sin⁡(θ)=sin⁡(90∘−θ)\sin(\theta)=\sin(90^\circ-\theta)
  3. sin⁡(θ)=cos⁡(90∘−θ)\sin(\theta)=\cos(90^\circ-\theta) (correct answer)
  4. sin⁡(θ)=cos⁡(−θ)\sin(\theta)=\cos(-\theta)
Explanation: This question tests understanding of the relationship between sine and cosine of complementary angles. On the unit circle, the coordinates at angle θ are (cos(θ), sin(θ)), and at the complementary angle 90° - θ, the coordinates exhibit a swap: (sin(θ), cos(θ)), demonstrating that sin(θ) = cos(90° - θ). The equation sin(θ) = cos(90° - θ) holds for all angles θ, meaning that if we know the sine of any angle, we automatically know the cosine of its complement (the angle that when added to θ gives 90°). Choice C is correct because it correctly states the cofunction relationship sin(θ) = cos(90° - θ) from the unit circle swap. Choice B claims sin(θ) = sin(90° - θ), using the same function for both angles, when the cofunction relationship requires switching from sine to cosine. To verify the complementary angle relationship on the unit circle, observe that as you move from angle θ to angle 90° - θ, the x and y coordinates swap positions, showing cos(θ) ↔ sin(90° - θ). Key to complementary angles: remember that two angles are complementary if they sum to 90° (or π/2), and the cofunction relationship sin(θ) = cos(90° - θ) allows you to convert between sine and cosine using the complement.

Question 15

Compare the two expressions sin⁡(20∘)\sin(20^\circ) and cos⁡(70∘)\cos(70^\circ). Given that 20∘20^\circ and 70∘70^\circ are complementary angles, which statement is correct?​

  1. sin⁡(20∘)=cos⁡(70∘)\sin(20^\circ)=\cos(70^\circ) (correct answer)
  2. sin⁡(20∘)=sin⁡(70∘)\sin(20^\circ)=\sin(70^\circ)
  3. sin⁡(20∘)=cos⁡(110∘)\sin(20^\circ)=\cos(110^\circ)
  4. sin⁡(20∘)=sec⁡(70∘)\sin(20^\circ)=\sec(70^\circ)
Explanation: This question tests understanding of the relationship between sine and cosine of complementary angles. Two angles are complementary if their sum equals 90° (or π/2 radians), and there is a fundamental relationship: sin(θ) = cos(90° - θ) and cos(θ) = sin(90° - θ) for any angle θ. Since 20° and 70° are complementary (they sum to 90°), we know that sin(20°) = cos(70°). Choice A is correct because it correctly applies sin(20°) = cos(70°) using the complementary relationship. Choice B claims sin(20°) = sin(70°), using the same function for both angles, when the cofunction relationship requires switching from sine to cosine. Key to complementary angles: remember that two angles are complementary if they sum to 90° (or π/2), and the cofunction relationship sin(θ) = cos(90° - θ) allows you to convert between sine and cosine using the complement. Don't confuse complementary (sum to 90°) with supplementary (sum to 180°)—for supplementary angles, there is no simple cofunction relationship between sine and cosine.

Question 16

In a right triangle ABCABC with a right angle at CC, the acute angles AA and BB are complementary, so A+B=90∘A+B=90^\circ. Using the complementary angle relationship, how does sin⁡(A)\sin(A) relate to cos⁡(B)\cos(B)?​

  1. sin⁡(A)=sin⁡(B)\sin(A)=\sin(B)
  2. sin⁡(A)=cos⁡(B)\sin(A)=\cos(B) (correct answer)
  3. sin⁡(A)=cos⁡(180∘−B)\sin(A)=\cos(180^\circ-B)
  4. sin⁡(A)=sec⁡(B)\sin(A)=\sec(B)
Explanation: This question tests understanding of the relationship between sine and cosine of complementary angles. In a right triangle, the two acute angles are always complementary (they sum to 90°), and by similarity, the side ratios reveal that the sine of one acute angle equals the cosine of the other acute angle. In a right triangle with acute angles A and B, where A + B = 90°, consider a side that is opposite to angle A (making sin(A) = opposite/hypotenuse). That same side is adjacent to angle B (making cos(B) = adjacent/hypotenuse = opposite/hypotenuse), showing sin(A) = cos(B). Choice B is correct because it correctly applies sin(A) = cos(B) based on the complementary relationship. Choice A claims sin(A) = sin(B), using the same function for both angles, when the cofunction relationship requires switching from sine to cosine. In right triangles, always remember that the two acute angles are complementary, so sin of one angle equals cos of the other angle—this is why cosine starts with 'co' (for complement).

Question 17

On the unit circle, the point at angle θ\theta is (cos⁡(θ),sin⁡(θ))(\cos(\theta),\sin(\theta)). If two angles are complementary, θ\theta and 90∘−θ90^\circ-\theta, which comparison correctly matches the cofunction relationship?

  1. cos⁡(θ)=cos⁡(90∘−θ)\cos(\theta)=\cos(90^\circ-\theta)
  2. sin⁡(θ)=sin⁡(90∘−θ)\sin(\theta)=\sin(90^\circ-\theta)
  3. sin⁡(θ)=cos⁡(90∘−θ)\sin(\theta)=\cos(90^\circ-\theta) (correct answer)
  4. sin⁡(θ)=cos⁡(−θ)\sin(\theta)=\cos(-\theta)
Explanation: This question tests understanding of the relationship between sine and cosine of complementary angles. On the unit circle, the coordinates at angle θ are (cos(θ), sin(θ)), and at the complementary angle 90° - θ, the coordinates exhibit a swap: (sin(θ), cos(θ)), demonstrating that sin(θ) = cos(90° - θ). The equation sin(θ) = cos(90° - θ) holds for all angles θ, meaning that if we know the sine of any angle, we automatically know the cosine of its complement (the angle that when added to θ gives 90°). Choice C is correct because it correctly states the cofunction relationship sin(θ) = cos(90° - θ) from the unit circle swap. Choice B claims sin(θ) = sin(90° - θ), using the same function for both angles, when the cofunction relationship requires switching from sine to cosine. To verify the complementary angle relationship on the unit circle, observe that as you move from angle θ to angle 90° - θ, the x and y coordinates swap positions, showing cos(θ) ↔ sin(90° - θ). Key to complementary angles: remember that two angles are complementary if they sum to 90° (or π/2), and the cofunction relationship sin(θ) = cos(90° - θ) allows you to convert between sine and cosine using the complement.

Question 18

In a right triangle ABCABC with a right angle at CC, the acute angles AA and BB are complementary, so A+B=90∘A+B=90^\circ. Using the complementary angle relationship, how does sin⁡(A)\sin(A) relate to cos⁡(B)\cos(B)?

  1. sin⁡(A)=sin⁡(B)\sin(A)=\sin(B)
  2. sin⁡(A)=cos⁡(B)\sin(A)=\cos(B) (correct answer)
  3. sin⁡(A)=cos⁡(180∘−B)\sin(A)=\cos(180^\circ-B)
  4. sin⁡(A)=sec⁡(B)\sin(A)=\sec(B)
Explanation: This question tests understanding of the relationship between sine and cosine of complementary angles. In a right triangle, the two acute angles are always complementary (they sum to 90°), and by similarity, the side ratios reveal that the sine of one acute angle equals the cosine of the other acute angle. In a right triangle with acute angles A and B, where A + B = 90°, consider a side that is opposite to angle A (making sin(A) = opposite/hypotenuse). That same side is adjacent to angle B (making cos(B) = adjacent/hypotenuse = opposite/hypotenuse), showing sin(A) = cos(B). Choice B is correct because it correctly applies sin(A) = cos(B) based on the complementary relationship. Choice A claims sin(A) = sin(B), using the same function for both angles, when the cofunction relationship requires switching from sine to cosine. In right triangles, always remember that the two acute angles are complementary, so sin of one angle equals cos of the other angle—this is why cosine starts with 'co' (for complement).

Question 19

Compare the two expressions sin⁡(20∘)\sin(20^\circ) and cos⁡(70∘)\cos(70^\circ). Given that 20∘20^\circ and 70∘70^\circ are complementary angles, which statement is correct?

  1. sin⁡(20∘)=cos⁡(70∘)\sin(20^\circ)=\cos(70^\circ) (correct answer)
  2. sin⁡(20∘)=sin⁡(70∘)\sin(20^\circ)=\sin(70^\circ)
  3. sin⁡(20∘)=cos⁡(110∘)\sin(20^\circ)=\cos(110^\circ)
  4. sin⁡(20∘)=sec⁡(70∘)\sin(20^\circ)=\sec(70^\circ)
Explanation: This question tests understanding of the relationship between sine and cosine of complementary angles. Two angles are complementary if their sum equals 90° (or π/2 radians), and there is a fundamental relationship: sin(θ) = cos(90° - θ) and cos(θ) = sin(90° - θ) for any angle θ. Since 20° and 70° are complementary (they sum to 90°), we know that sin(20°) = cos(70°). Choice A is correct because it correctly applies sin(20°) = cos(70°) using the complementary relationship. Choice B claims sin(20°) = sin(70°), using the same function for both angles, when the cofunction relationship requires switching from sine to cosine. Key to complementary angles: remember that two angles are complementary if they sum to 90° (or π/2), and the cofunction relationship sin(θ) = cos(90° - θ) allows you to convert between sine and cosine using the complement. Don't confuse complementary (sum to 90°) with supplementary (sum to 180°)—for supplementary angles, there is no simple cofunction relationship between sine and cosine.

Question 20

Verify the cofunction relationship using complementary angles: which equation is true?

  1. cos⁡(40∘)=sin⁡(50∘)\cos(40^\circ)=\sin(50^\circ) (correct answer)
  2. cos⁡(40∘)=cos⁡(50∘)\cos(40^\circ)=\cos(50^\circ)
  3. cos⁡(40∘)=sin⁡(140∘)\cos(40^\circ)=\sin(140^\circ)
  4. cos⁡(40∘)=sin⁡(−50∘)\cos(40^\circ)=\sin(-50^\circ)
Explanation: This question tests understanding of the relationship between sine and cosine of complementary angles. Two angles are complementary if their sum equals 90° (or π/2 radians), and there is a fundamental relationship: sin(θ) = cos(90° - θ) and cos(θ) = sin(90° - θ) for any angle θ. Since 40° and 50° are complementary (they sum to 90°), we know that cos(40°) = sin(90° - 40°) = sin(50°). Choice A is correct because it correctly states that cos(40°) = sin(50°) when 40° and 50° are complementary angles. Choice B incorrectly claims cos(40°) = cos(50°), which would only be true if the angles were equal, but 40° ≠ 50°. Key to complementary angles: remember that two angles are complementary if they sum to 90° (or π/2), and the cofunction relationship sin(θ) = cos(90° - θ) allows you to convert between sine and cosine using the complement.