Sine and Cosine of Complementary Angles - Precalculus

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

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

2. What is the definition of $sin(\theta)$ in a right triangle for acute $\theta$?

   $sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}$. Sine is the ratio of the opposite side to the hypotenuse.

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

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

4. What is the definition of $csc(\theta)$ in a right triangle for acute $\theta$?

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

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

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

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

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

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

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

8. What is the definition of $\frac{\text{hypotenuse}}{\text{adjacent}}$ for acute angle $\theta$?

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

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

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

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

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

11. What relationship between right triangles justifies trig ratios depending only on angle $\theta$?

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

12. What is the definition of $cot(\theta)$ in a right triangle for acute $\theta$?

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

13. What is the definition of $tan(\theta)$ in a right triangle for acute $\theta$?

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

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

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

15. What is the reciprocal identity for tangent and cotangent?

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

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

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

17. What is the definition of $cos(\theta)$ in a right triangle for acute $\theta$?

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

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

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

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

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

20. What is the definition of $\frac{\text{hypotenuse}}{\text{opposite}}$ for acute angle $\theta$?

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

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

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

22. What is the definition of $\frac{\text{adjacent}}{\text{hypotenuse}}$ for acute angle $\theta$?

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

23. Identify the side called opposite relative to an acute angle $\theta$ in a right triangle.

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

24. What is the definition of $sec(\theta)$ in a right triangle for acute $\theta$?

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

25. What is the definition of $\tan(\theta)$ in a right triangle for acute $\theta$?

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

26. Identify the side called adjacent relative to an acute angle $\theta$ in a right triangle.

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

27. What is the reciprocal identity for sine and cosecant?

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

28. What is the definition of $\frac{\text{opposite}}{\text{hypotenuse}}$ for acute angle $\theta$?

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

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

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

30. What is the definition of $\frac{\text{adjacent}}{\text{opposite}}$ for acute angle $\theta$?

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

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

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

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

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

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

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

34. What is the reciprocal identity for cosine and secant?

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

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

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

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

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

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

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

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