Precalculus Flashcards: Sine And Cosine Of Complementary Angles

What this deck covers

This deck focuses on Sine And Cosine Of Complementary Angles, giving you a quick way to review the definitions, rules, and examples that matter most for Precalculus.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: What does similarity imply about ratios of corresponding sides in similar triangles?

Answer: Corresponding side ratios are equal (a constant scale factor). Similar triangles have proportional sides with the same ratio.

Flashcard 2: What is the definition of sin( heta) in a right triangle for acute  heta?

Answer: sin( heta)=\frac{\text{opposite}}{\text{hypotenuse}}. Sine is the ratio of the opposite side to the hypotenuse.

Flashcard 3: Identify the value of $\tan(\theta)$ if opposite $=7$ and adjacent $=4$ for acute $\theta$.

Answer: $\tan(\theta)=\frac{7}{4}$. Apply tangent definition: opposite over adjacent.

Flashcard 4: What is the definition of csc( heta) in a right triangle for acute  heta?

Answer: csc(\theta)=\frac{\text{hypotenuse}}{\text{opposite}}. Cosecant is the reciprocal of sine.

Flashcard 5: Identify the side used as the hypotenuse in a right triangle.

Answer: The side opposite the $90^\circ$ angle (the longest side). The hypotenuse is always opposite the right angle.

Flashcard 6: What is the complementary-angle identity for tangent in a right triangle?

Answer: $\tan(\theta)=\cot(90^\circ-\theta)$. Complementary angles have reciprocal tan/cot values in right triangles.

Flashcard 7: What is the complementary-angle identity for cosine and sine in a right triangle?

Answer: cos(\theta)=sin(90^\circ-\theta). Complementary angles in right triangles interchange sine and cosine.

Flashcard 8: What is the definition of rac{\text{hypotenuse}}{\text{adjacent}} for acute angle $heta$?

Answer: $\sec(\theta)$. Secant is the reciprocal of cosine: hypotenuse over adjacent.

Flashcard 9: Find cos(\theta) if adjacent $=12$ and hypotenuse $=13$.

Answer: cos(\theta)=\frac{12}{13}. Apply the cosine ratio formula directly.

Flashcard 10: What is $\cos(\theta)$ if $\sin(\theta)=\frac{5}{13}$ and $\theta$ is acute?

Answer: $\cos(\theta)=\frac{12}{13}$. Use $\sin^2(\theta)+\cos^2(\theta)=1$ with $\sin(\theta)=\frac{5}{13}$.

Flashcard 11: What relationship between right triangles justifies trig ratios depending only on angle $heta$?

Answer: Right triangles with the same acute angle are similar. AA similarity ensures all right triangles with same acute angle have equal trig ratios.

Flashcard 12: What is the definition of cot( heta) in a right triangle for acute  heta?

Answer: cot(\theta)=\frac{\text{adjacent}}{\text{opposite}}. Cotangent is the reciprocal of tangent.

Flashcard 13: What is the definition of tan( heta) in a right triangle for acute  heta?

Answer: tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}. Tangent is the ratio of opposite to adjacent sides.

Flashcard 14: What does AA similarity mean for triangles (state the criterion)?

Answer: If two angles match, the triangles are similar. AA (Angle-Angle) similarity requires only two pairs of equal angles.

Flashcard 15: What is the reciprocal identity for tangent and cotangent?

Answer: $\cot(\theta)=\frac{1}{\tan(\theta)}$. Cotangent and tangent are reciprocals of each other.

Flashcard 16: What is the complementary-angle identity for sine in a right triangle?

Answer: $\sin(\theta)=\cos(90^\circ-\theta)$. In a right triangle, complementary angles swap opposite and adjacent sides.

Flashcard 17: What is the definition of cos( heta) in a right triangle for acute  heta?

Answer: cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}. Cosine is the ratio of the adjacent side to the hypotenuse.

Flashcard 18: Identify the value of $\sin(\theta)$ if opposite $=3$ and hypotenuse $=5$ for acute $\theta$.

Answer: $\sin(\theta)=\frac{3}{5}$. Apply sine definition: opposite over hypotenuse.

Flashcard 19: Find tan(\theta) if opposite $=8$ and adjacent $=6$.

Answer: tan(\theta)=\frac{8}{6}=\frac{4}{3}. Apply the tangent ratio formula and simplify.

Flashcard 20: What is the definition of rac{\text{hypotenuse}}{\text{opposite}} for acute angle $heta$?

Answer: $\csc(\theta)$. Cosecant is the reciprocal of sine: hypotenuse over opposite.

Flashcard 21: Find cos((90^\circ-\theta)) if sin(\theta)=\frac{7}{9}.

Answer: cos((90^\circ-\theta))=\frac{7}{9}. Use the identity $\cos(90°-\theta) = \sin(\theta)$.

Flashcard 22: What is the definition of rac{\text{adjacent}}{\text{hypotenuse}} for acute angle $heta$?

Answer: $\cos(\theta)$. Cosine is defined as adjacent over hypotenuse in a right triangle.

Flashcard 23: Identify the side called opposite relative to an acute angle  heta in a right triangle.

Answer: The side across from \theta. The opposite side doesn't touch the angle $\theta$.

Flashcard 24: What is the definition of sec( heta) in a right triangle for acute  heta?

Answer: sec(\theta)=\frac{\text{hypotenuse}}{\text{adjacent}}. Secant is the reciprocal of cosine.

Flashcard 25: What is the definition of $an( heta)$ in a right triangle for acute $heta$?

Answer: $\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}$. Tangent is the ratio of the opposite side to the adjacent side.

Flashcard 26: Identify the side called adjacent relative to an acute angle  heta in a right triangle.

Answer: The non-hypotenuse side that touches \theta. Adjacent means next to the angle, excluding the hypotenuse.

Flashcard 27: What is the reciprocal identity for sine and cosecant?

Answer: $\csc(\theta)=\frac{1}{\sin(\theta)}$. Cosecant and sine are reciprocals of each other.

Flashcard 28: What is the definition of rac{\text{opposite}}{\text{hypotenuse}} for acute angle $heta$?

Answer: $\sin(\theta)$. Sine is defined as opposite over hypotenuse in a right triangle.

Flashcard 29: Identify the value of $\cos(\theta)$ if adjacent $=12$ and hypotenuse $=13$ for acute $\theta$.

Answer: $\cos(\theta)=\frac{12}{13}$. Apply cosine definition: adjacent over hypotenuse.

Flashcard 30: What is the definition of rac{\text{adjacent}}{\text{opposite}} for acute angle $heta$?

Answer: $\cot(\theta)$. Cotangent is the reciprocal of tangent: adjacent over opposite.

Flashcard 31: What is the complementary-angle identity relating tangent and cotangent?

Answer: tan(\theta)=cot(90^\circ-\theta). Complementary angles swap tangent and cotangent values.

Flashcard 32: What is the definition of similar triangles in terms of angles and side ratios?

Answer: Corresponding angles equal and corresponding sides proportional. Similar triangles have matching angles and proportional corresponding sides.

Flashcard 33: What is the complementary-angle identity for sine and cosine in a right triangle?

Answer: sin(\theta)=cos(90^\circ-\theta). In a right triangle, complementary angles swap opposite and adjacent sides.

Flashcard 34: What is the reciprocal identity for cosine and secant?

Answer: $\sec(\theta)=\frac{1}{\cos(\theta)}$. Secant and cosine are reciprocals of each other.

Flashcard 35: What is the complementary-angle identity relating secant and cosecant?

Answer: sec(\theta)=csc(90^\circ-\theta). Complementary angles swap secant and cosecant values.

Flashcard 36: Find tan((90^\circ-\theta)) if tan(\theta)=\frac{3}{5}.

Answer: tan((90^\circ-\theta))=\frac{5}{3}. Use $\tan(90°-\theta) = \cot(\theta) = \frac{1}{\tan(\theta)}$.

Flashcard 37: Find sin(\theta) if opposite $=9$ and hypotenuse $=15$.

Answer: sin(\theta)=\frac{9}{15}=\frac{3}{5}. Apply the sine ratio formula and simplify the fraction.