Sine and Cosine Relationship of Complementary Angles: CCSS.Math.Content.HSG-SRT.C.7 - Geometry

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Question

True or False: The sine of any acute angle is equal to the cosine of its complement.

Answer

Consider the triangle below

First we must understand the definition of complementary angles. Complementary angles are a pair of angles that add up to 90 degrees. Looking at the triangle above, we see that $\angle BAC$ and $\angle ACB$ add up to 90, so they are complementary.

Recall that the definition of sine is $sin = \frac{opposite}{hypotenuse}$ and cosine is $cos = \frac{adjacent}{hypotenuse}$ .

Next, list out the sine and cosine of $\angle BAC$ and $\angle ACB$ .

$sin(\angle BAC) = \frac{10.83}{11.39}$

$sin(\angle ACB) = \frac{3.52}{11.39}$

$cos(\angle BAC) = \frac{3.52}{11.39}$

$cos(\angle ACB) = \frac{10.83}{11.39}$

Notice that $sin(\angle BAC) = \frac{10.83}{11.39} = cos(\angle ACB)$

And $sin(\angle ACB) = \frac{3.52}{11.39} = cos(\angle BAC)$

We have just shown that the sine of the two angles is equal to the cosines of their complements. This is true for any acute angles.

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Question

You have two complementary angles, $\angle A$ and $\angle B$ . You know that $\angle A = 30$ . Using your sine and cosine relationships, what is $\angle B$ in degrees?

Answer

You know that $\angle A = 30$ . We use this information to set up our work in the following way:

$sin(\angle A) = cos(\angle B)$ $sin(\angle A) = cos(\angle B)$

$\implies sin(30) = cos(\angle B)$

$\implies \frac{1}{2} = cos(\angle B)$

$\implies cos^{-1}(\frac{1}{2}) = \angle B$

$\implies 60 = \angle B$

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Question

You have two complementary angles, $\angle A = 2x + 5$ and $\angle B = 4x - 12$ . Solve for , round to the second decimal place.

Answer

To solve for , we need to use the definition of complementary. Both of these angles should add up to be 90, so we can set up this problem by setting the sum of the two angles equal to 90.

$\angle A + \angle B = 90$

$\implies (2x + 5) + (4x -12) = 90$

$\implies 6x - 7 = 90$

$\implies 6x = 97$

$\implies x = 16.17$

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Question

Consider the equilateral triangle below. Find the height of the triangle using the sine cosine relationship of complementary angles.

Answer

We begin by noting that this triangle is an equilateral triangle as stated in the problem. This means that all sides are congruent and all angles are congruent. Since a triangle’s angles must add up to 180 degrees, we can deduce that each of the three angles is 60 degrees. By drawing a vertical auxiliary line down from vertex A perpendicular to the base of the triangle we are evenly dividing the equilateral triangle into two right triangles. We can update the triangle as such:

We also know that each of the three angles of the equilateral triangle are equal to 60 degrees. Since the auxiliary line splits the equilateral evenly into two right triangles, we know that $\angle BAD = \angle CAD = 30$ and that $\bar{BD} = \bar{CD} = 1$ . Again, we can update our triangle:

Now we can begin to use our sine and cosine relationships of complementary angles to solve for . Recall the following:

$sin(\theta) = \frac{opposite}{hypotenuse}$

$cos(\theta) = \frac{adjacent}{hypotenuse}$

Since we can use either one to solve for this problem.

$sin(60) = \frac{h}{2}$

$\frac{\sqrt{3}}{2} = \frac{h}{2}$

$\sqrt{3} = h$

or

$cos(30) = \frac{h}{2}$

$\frac{\sqrt{3}}{2} = \frac{h}{2}$

$\sqrt{3} = h$

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Question

True or False: The sine and cosine relationship of complementary angles is only true for the acute angles of a right triangle.

Answer

This relationship is true for any pair of angles that are complementary (i.e. any two angles that add up to 90). We can see this in obtuse triangles that have two acute angles that add up to 90, we can see it in acute triangles, and equilateral triangles as well. Beyond triangles this relationship is also true.

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Question

Use the sine and cosine relationship of complementary angles to find the diagonal of the square. (DO NOT use Pythagorean Theorem).

Answer

First, we need to think about what it means to be a square. Squares have 4 congruent sides and 4 congruent angles that must add up to 360. So each angle must be 90-degree angles. The diagonal of the square cut two of these 90-degree angles in symmetrical halves to form two right triangles within the square. Now we have two identical right triangles with legs of length 5 and angles of 90, 45, 45.

Recall what we know about our sine and cosine relationships:

$sin(\theta) = cos(90-\theta)$

So we can use either sine or cosine to solve for the hypotenuse. We will first use cosine:

$cos(45) = \frac{adjacent}{hypotenuse}$ $cos(45) = \frac{adjacent}{hypotenuse}$

$cos(45) = \frac{5}{L}$

$\frac{1}{\sqrt{2}} = \frac{5}{L}$

$L = 5\sqrt{2}$

We can also use sine since their relationship makes them equivalent

$sin(45) = \frac{adjacent}{hypotenuse}$

$sin(45) = \frac{5}{L}$

$\frac{1}{\sqrt{2}} = \frac{5}{L}$

$L = 5\sqrt{2}$

And now we can confirm using the Pythagorean Theorem to check our work:

$c = \sqrt{50}$

$c = \sqrt{25*2}$

$c = 5\sqrt(2)$

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Question

Use the sine and cosine relationship of complementary angles to solve for .

Answer

Recall our sine and cosine relationship of complementary angles. The sine of any acute angle is equal to the cosine of its complement:

$sin(\theta) = cos(90 - \theta)$

We know that $sin(\theta) = \frac{opposite}{hypotenuse}$ and $cos(\theta) = \frac{adjacent}{hypotenuse}$ .

We can use this information to set up the following equation to solve for .

$sin(\theta) = cos(90 - \theta)$

$\implies sin(32) = cos(58)$

$\implies \frac{2 + x}{38 - x} = 0.53$ ()

$\implies 2 + x = 20.14 - 0.53x$ (multiply both sides by )

$\implies 1.53x = 18.14$

$\implies x = 11.87$

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Question

You have two complementary angles, $\angle A$ and $\angle B$ . You know that $\angle A = 65 - 2x$ and $\angle B = 24 + x$ . Using your sine and cosine relationships, what is ?

Answer

We will use the definition of complementary to solve for . $\angle A$ and $\angle B$ must sum up to be 90 by the definition of complementary angles. We are able to set up the following equation:

$\angle A + \angle B = 90$

$\implies (65 - 2x) + (24 + x) = 90$

$\implies 89 +x = 90$ (combine like terms)

$\implies x = 1$

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Question

Consider the following figure. $\bar{AB}$ and $\bar{CD}$ are parallel to each other. $\bar{EF}$ and $\bar{GH}$ are parallel to each other. Is there enough information to solve for using sine and cosine relationship of complementary angles? If so, what is the value of ?

Answer

We can deduce that $\angle CAD$ and $\angle DCA$ are complementary angles because triangle is a right triangle. We have enough information from this figure to find $\angle CAD$ . Once we have this measurement we will be able to use the sine cosine relationship of complementary angles to solve for .

Since $\bar{BD}$ is a straight line segment, $\angle BAD + \angle CAD = 180$ . Plugging in our values we can find

$\angle BAD + \angle CAD = 180$

$\angle BAD + \angle CAD = 180$

$\implies \angle CAD = 55$

We could just use our knowledge that a triangle must add up to 180 degrees to solve for but that is not what the question is asking. We must now use the fact that $sin(\theta) = cos(90-\theta$ ). We can set up the following equation to solve for .

$\implies 0.82 = cos(x)$

$\implies cos^{-1}(0.82) = x$

$\implies 35 = x$

Compare your answer with the correct one above

Question

$\angle A$ and $\angle B$ are complementary angles. If what is $\angle B$ ? Use the sine cosine relationship of complementary angles to solve this problem.

Answer

Using our relationship we know that $sin(\angle A) = cos(\angle B)$ . We can set up the following equation to solve for $\angle B$

$sin(\angle A) = cos(\angle B)$

$\implies sin(80) = cos(\angle B)$

$\implies 0.9848 = cos(\angle B)$

$\implies cos^{-1}(0.9848) = \angle B$

$\implies 10 = \angle B$

We can double check our work using the definition of complementary angles. That means that $\angle A + \angle B = 90$ . We know that our answer must be correct because .

Compare your answer with the correct one above

Question

True or False: The sine of any acute angle is equal to the cosine of its complement.

Answer

Consider the triangle below

First we must understand the definition of complementary angles. Complementary angles are a pair of angles that add up to 90 degrees. Looking at the triangle above, we see that $\angle BAC$ and $\angle ACB$ add up to 90, so they are complementary.

Recall that the definition of sine is $sin = \frac{opposite}{hypotenuse}$ and cosine is $cos = \frac{adjacent}{hypotenuse}$ .

Next, list out the sine and cosine of $\angle BAC$ and $\angle ACB$ .

$sin(\angle BAC) = \frac{10.83}{11.39}$

$sin(\angle ACB) = \frac{3.52}{11.39}$

$cos(\angle BAC) = \frac{3.52}{11.39}$

$cos(\angle ACB) = \frac{10.83}{11.39}$

Notice that $sin(\angle BAC) = \frac{10.83}{11.39} = cos(\angle ACB)$

And $sin(\angle ACB) = \frac{3.52}{11.39} = cos(\angle BAC)$

We have just shown that the sine of the two angles is equal to the cosines of their complements. This is true for any acute angles.

Compare your answer with the correct one above

Question

You have two complementary angles, $\angle A$ and $\angle B$ . You know that $\angle A = 30$ . Using your sine and cosine relationships, what is $\angle B$ in degrees?

Answer

You know that $\angle A = 30$ . We use this information to set up our work in the following way:

$sin(\angle A) = cos(\angle B)$ $sin(\angle A) = cos(\angle B)$

$\implies sin(30) = cos(\angle B)$

$\implies \frac{1}{2} = cos(\angle B)$

$\implies cos^{-1}(\frac{1}{2}) = \angle B$

$\implies 60 = \angle B$

Compare your answer with the correct one above

Question

You have two complementary angles, $\angle A = 2x + 5$ and $\angle B = 4x - 12$ . Solve for , round to the second decimal place.

Answer

To solve for , we need to use the definition of complementary. Both of these angles should add up to be 90, so we can set up this problem by setting the sum of the two angles equal to 90.

$\angle A + \angle B = 90$

$\implies (2x + 5) + (4x -12) = 90$

$\implies 6x - 7 = 90$

$\implies 6x = 97$

$\implies x = 16.17$

Compare your answer with the correct one above

Question

Consider the equilateral triangle below. Find the height of the triangle using the sine cosine relationship of complementary angles.

Answer

We begin by noting that this triangle is an equilateral triangle as stated in the problem. This means that all sides are congruent and all angles are congruent. Since a triangle’s angles must add up to 180 degrees, we can deduce that each of the three angles is 60 degrees. By drawing a vertical auxiliary line down from vertex A perpendicular to the base of the triangle we are evenly dividing the equilateral triangle into two right triangles. We can update the triangle as such:

We also know that each of the three angles of the equilateral triangle are equal to 60 degrees. Since the auxiliary line splits the equilateral evenly into two right triangles, we know that $\angle BAD = \angle CAD = 30$ and that $\bar{BD} = \bar{CD} = 1$ . Again, we can update our triangle:

Now we can begin to use our sine and cosine relationships of complementary angles to solve for . Recall the following:

$sin(\theta) = \frac{opposite}{hypotenuse}$

$cos(\theta) = \frac{adjacent}{hypotenuse}$

Since we can use either one to solve for this problem.

$sin(60) = \frac{h}{2}$

$\frac{\sqrt{3}}{2} = \frac{h}{2}$

$\sqrt{3} = h$

or

$cos(30) = \frac{h}{2}$

$\frac{\sqrt{3}}{2} = \frac{h}{2}$

$\sqrt{3} = h$

Compare your answer with the correct one above

Question

True or False: The sine and cosine relationship of complementary angles is only true for the acute angles of a right triangle.

Answer

This relationship is true for any pair of angles that are complementary (i.e. any two angles that add up to 90). We can see this in obtuse triangles that have two acute angles that add up to 90, we can see it in acute triangles, and equilateral triangles as well. Beyond triangles this relationship is also true.

Compare your answer with the correct one above

Question

Use the sine and cosine relationship of complementary angles to find the diagonal of the square. (DO NOT use Pythagorean Theorem).

Answer

First, we need to think about what it means to be a square. Squares have 4 congruent sides and 4 congruent angles that must add up to 360. So each angle must be 90-degree angles. The diagonal of the square cut two of these 90-degree angles in symmetrical halves to form two right triangles within the square. Now we have two identical right triangles with legs of length 5 and angles of 90, 45, 45.

Recall what we know about our sine and cosine relationships:

$sin(\theta) = cos(90-\theta)$

So we can use either sine or cosine to solve for the hypotenuse. We will first use cosine:

$cos(45) = \frac{adjacent}{hypotenuse}$ $cos(45) = \frac{adjacent}{hypotenuse}$

$cos(45) = \frac{5}{L}$

$\frac{1}{\sqrt{2}} = \frac{5}{L}$

$L = 5\sqrt{2}$

We can also use sine since their relationship makes them equivalent

$sin(45) = \frac{adjacent}{hypotenuse}$

$sin(45) = \frac{5}{L}$

$\frac{1}{\sqrt{2}} = \frac{5}{L}$

$L = 5\sqrt{2}$

And now we can confirm using the Pythagorean Theorem to check our work:

$c = \sqrt{50}$

$c = \sqrt{25*2}$

$c = 5\sqrt(2)$

Compare your answer with the correct one above

Question

Use the sine and cosine relationship of complementary angles to solve for .

Answer

Recall our sine and cosine relationship of complementary angles. The sine of any acute angle is equal to the cosine of its complement:

$sin(\theta) = cos(90 - \theta)$

We know that $sin(\theta) = \frac{opposite}{hypotenuse}$ and $cos(\theta) = \frac{adjacent}{hypotenuse}$ .

We can use this information to set up the following equation to solve for .

$sin(\theta) = cos(90 - \theta)$

$\implies sin(32) = cos(58)$

$\implies \frac{2 + x}{38 - x} = 0.53$ ()

$\implies 2 + x = 20.14 - 0.53x$ (multiply both sides by )

$\implies 1.53x = 18.14$

$\implies x = 11.87$

Compare your answer with the correct one above

Question

You have two complementary angles, $\angle A$ and $\angle B$ . You know that $\angle A = 65 - 2x$ and $\angle B = 24 + x$ . Using your sine and cosine relationships, what is ?

Answer

We will use the definition of complementary to solve for . $\angle A$ and $\angle B$ must sum up to be 90 by the definition of complementary angles. We are able to set up the following equation:

$\angle A + \angle B = 90$

$\implies (65 - 2x) + (24 + x) = 90$

$\implies 89 +x = 90$ (combine like terms)

$\implies x = 1$

Compare your answer with the correct one above

Question

Consider the following figure. $\bar{AB}$ and $\bar{CD}$ are parallel to each other. $\bar{EF}$ and $\bar{GH}$ are parallel to each other. Is there enough information to solve for using sine and cosine relationship of complementary angles? If so, what is the value of ?

Answer

We can deduce that $\angle CAD$ and $\angle DCA$ are complementary angles because triangle is a right triangle. We have enough information from this figure to find $\angle CAD$ . Once we have this measurement we will be able to use the sine cosine relationship of complementary angles to solve for .

Since $\bar{BD}$ is a straight line segment, $\angle BAD + \angle CAD = 180$ . Plugging in our values we can find

$\angle BAD + \angle CAD = 180$

$\angle BAD + \angle CAD = 180$

$\implies \angle CAD = 55$

We could just use our knowledge that a triangle must add up to 180 degrees to solve for but that is not what the question is asking. We must now use the fact that $sin(\theta) = cos(90-\theta$ ). We can set up the following equation to solve for .

$\implies 0.82 = cos(x)$

$\implies cos^{-1}(0.82) = x$

$\implies 35 = x$

Compare your answer with the correct one above

Question

$\angle A$ and $\angle B$ are complementary angles. If what is $\angle B$ ? Use the sine cosine relationship of complementary angles to solve this problem.

Answer

Using our relationship we know that $sin(\angle A) = cos(\angle B)$ . We can set up the following equation to solve for $\angle B$

$sin(\angle A) = cos(\angle B)$

$\implies sin(80) = cos(\angle B)$

$\implies 0.9848 = cos(\angle B)$

$\implies cos^{-1}(0.9848) = \angle B$

$\implies 10 = \angle B$

We can double check our work using the definition of complementary angles. That means that $\angle A + \angle B = 90$ . We know that our answer must be correct because .

Compare your answer with the correct one above

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