Solving Right Triangles: Pythagorean Theorem, Trigonometry - Precalculus

1. If $\cos(\theta)=0.35$ and $\theta$ is acute, what is $\sin(90^\circ-\theta)$?

   $\sin(90^\circ-\theta)=0.35$. By the complementary angle identity, $\sin(90°-\theta)=\cos(\theta)$.

2. What is $\sin(90^\circ-\theta)$ equal to in terms of $\theta$?

   $\sin(90^\circ-\theta)=\cos(\theta)$. Since $\theta$ and $90°-\theta$ are complementary, sine becomes cosine.

3. Which option correctly rewrites $\cos(20^\circ)$ using sine of a complementary angle?

   $\cos(20^\circ)=\sin(70^\circ)$. Since $20°+70°=90°$, they are complementary angles.

4. Find $\sin(30^\circ)$ using a complementary-angle identity and a known cosine value.

   $\sin(30^\circ)=\cos(60^\circ)=\frac{1}{2}$. Since $30°$ and $60°$ are complementary, $\sin(30°)=\cos(60°)$.

5. In a right triangle, which ratio equals $\sin(A)$ in terms of $\cos(B)$ when $A$ and $B$ are complementary?

   $\sin(A)=\cos(B)$. In a right triangle, opposite/hypotenuse for one angle equals adjacent/hypotenuse for the other.

6. If $\sin(\theta)=0.8$ and $\theta$ is acute, what is $\cos(90^\circ-\theta)$?

   $0.8$. By the cofunction identity, $\cos(90°-\theta)=\sin(\theta)$.

7. In a right triangle with acute complementary angles $A$ and $B$, what is $\cos(A)$ equal to?

   $\cos(A)=\sin(B)$. Since $A+B=90°$, cosine of one angle equals sine of its complement.

8. Find and correct the error: $\sin(\theta)=\cos(90^\circ+\theta)$ for acute $\theta$.

   Correct: $\sin(\theta)=\cos(90^\circ-\theta)$. The error uses addition instead of subtraction for complementary angles.

9. What is $\sin(90^\circ-\theta)$ in terms of $\cos(\theta)$?

   $\sin(90^\circ-\theta)=\cos(\theta)$. The sine of an angle's complement equals the angle's cosine.

10. In a right triangle with acute complementary angles $A$ and $B$, what is $\sin(A)$ equal to?

    $\sin(A)=\cos(B)$. Since $A+B=90°$, sine of one angle equals cosine of its complement.

11. Find the complement of $17^\circ$ (the angle that adds to $17^\circ$ to make $90^\circ$).

    $73^\circ$. Complementary angles sum to $90°$, so $90°-17°=73°$.

12. Find $\sin(60^\circ)$ using a complementary-angle identity and a known cosine value.

    $\sin(60^\circ)=\cos(30^\circ)=\frac{\sqrt{3}}{2}$. Since $60°$ and $30°$ are complementary, $\sin(60°)=\cos(30°)$.

13. Evaluate $\cos(90^\circ-35^\circ)$ by rewriting it as a sine function.

    $\sin(35^\circ)$. $90°-35°=55°$, and $\cos(55°)=\sin(35°)$ by the cofunction identity.

14. State the complementary-angle identity that rewrites $\sin(\theta)$ using cosine.

    $\sin(\theta)=\cos(90^\circ-\theta)$. Complementary angles sum to $90°$, so sine of one equals cosine of the other.

15. In a right triangle, which ratio equals $\cos(A)$ in terms of $\sin(B)$ when $A$ and $B$ are complementary?

    $\cos(A)=\sin(B)$. In a right triangle, adjacent/hypotenuse for one angle equals opposite/hypotenuse for the other.

16. Identify the cofunction pair: which function equals $\sin(\theta)$ for complementary angles?

    $\cos(90^\circ-\theta)$. Sine and cosine are cofunctions for complementary angles.

17. If $\cos(\theta)=0.6$ and $\theta$ is acute, what is $\sin(90^\circ-\theta)$?

    $0.6$. By the cofunction identity, $\sin(90°-\theta)=\cos(\theta)$.

18. State the complementary-angle identity that relates $\cos(\theta)$ and $\sin(90^\circ-\theta)$.

    $\cos(\theta)=\sin(90^\circ-\theta)$. Cosine of an angle equals sine of its complement.

19. Find and correct the identity error: $\sin(\theta)=\sin(90^\circ-\theta)$.

    Correct: $\sin(\theta)=\cos(90^\circ-\theta)$. The error uses sine on both sides; the correct identity uses cosine on the right.

20. State the complementary-angle identity that relates $\sin(\theta)$ and $\cos(90^\circ-\theta)$.

    $\sin(\theta)=\cos(90^\circ-\theta)$. Sine of an angle equals cosine of its complement.

21. Identify the cofunction pair: which function equals $\cos(\theta)$ for complementary angles?

    $\sin(90^\circ-\theta)$. Cosine and sine are cofunctions for complementary angles.

22. What is $\cos(90^\circ-\theta)$ in terms of $\sin(\theta)$?

    $\cos(90^\circ-\theta)=\sin(\theta)$. The cosine of an angle's complement equals the angle's sine.

23. What is $\cos(30^\circ)$ using a complementary-angle relationship with $\sin(60^\circ)$?

    $\cos(30^\circ)=\sin(60^\circ)$. $30°$ and $60°$ are complementary, so their cosine and sine are equal.

24. Identify the missing expression to make a true identity: $\cos(\theta)=\sin(\,\underline{\hspace{1.2cm}}\,)$.

    $90^\circ-\theta$. To make $\cos(\theta)=\sin(?)$, the angle must be $\theta$'s complement.

25. State the complementary-angle identity that rewrites $\cos(\theta)$ using sine.

    $\cos(\theta)=\sin(90^\circ-\theta)$. Complementary angles sum to $90°$, so cosine of one equals sine of the other.

26. What is $\cos(90^\circ-\theta)$ equal to in terms of $\theta$?

    $\cos(90^\circ-\theta)=\sin(\theta)$. Since $\theta$ and $90°-\theta$ are complementary, cosine becomes sine.

27. Evaluate $\sin(90^\circ-20^\circ)$ by rewriting it as a cosine function.

    $\cos(20^\circ)$. $90°-20°=70°$, and $\sin(70°)=\cos(20°)$ by the cofunction identity.

28. What is $\sin(30^\circ)$ using a complementary-angle relationship with $\cos(60^\circ)$?

    $\sin(30^\circ)=\cos(60^\circ)$. $30°$ and $60°$ are complementary, so their sine and cosine are equal.

29. Find $\cos(30^\circ)$ using a complementary-angle identity and a known sine value.

    $\cos(30^\circ)=\sin(60^\circ)=\frac{\sqrt{3}}{2}$. Since $30°$ and $60°$ are complementary, $\cos(30°)=\sin(60°)$.

30. If $A$ and $B$ are complementary acute angles and $\cos(A)=\frac{12}{13}$, what is $\sin(B)$?

    $\sin(B)=\frac{12}{13}$. Since $A$ and $B$ are complementary, $\cos(A)=\sin(B)$.

31. In a right triangle, if acute angles are $A$ and $B$, what is the relationship between $A$ and $B$?

    $A+B=90^\circ$. The two acute angles in a right triangle are complementary.

32. Identify the missing expression to make a true identity: $\sin(\theta)=\cos(\,\underline{\hspace{1.2cm}}\,)$.

    $90^\circ-\theta$. To make $\sin(\theta)=\cos(?)$, the angle must be $\theta$'s complement.

33. If $A$ and $B$ are complementary acute angles and $\sin(A)=\frac{3}{5}$, what is $\cos(B)$?

    $\cos(B)=\frac{3}{5}$. Since $A$ and $B$ are complementary, $\sin(A)=\cos(B)$.

34. Find $\cos(60^\circ)$ using a complementary-angle identity and a known sine value.

    $\cos(60^\circ)=\sin(30^\circ)=\frac{1}{2}$. Since $60°$ and $30°$ are complementary, $\cos(60°)=\sin(30°)$.

35. If $\theta$ is acute, what is the complement of $\theta$ written in degrees?

    $90^\circ-\theta$. The complement is the angle that adds to $\theta$ to make $90°$.

36. Given $\cos(\beta)=\frac{12}{13}$ and $\beta$ is acute, find $\sin(90^\circ-\beta)$.

    $\frac{12}{13}$. Using $\sin(90°-\beta)=\cos(\beta)$.

37. Identify the relationship between acute complementary angles $A$ and $B$ in a right triangle.

    $A+B=90^\circ$. Complementary angles sum to $90°$ by definition.

38. Find $\sin(45^\circ)$ by rewriting it as a cosine of a complementary angle.

    $\sin(45^\circ)=\cos(45^\circ)=\frac{\sqrt{2}}{2}$. $45°$ is its own complement, so $\sin(45°)=\cos(45°)$.

39. Given $\sin(\alpha)=\frac{3}{5}$ and $\alpha$ is acute, find $\cos(90^\circ-\alpha)$.

    $\frac{3}{5}$. Using $\cos(90°-\alpha)=\sin(\alpha)$.

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