Equivalent Representations of Trigonometric Functions - AP Precalculus

1. State the identity for $1 + \tan^2(\theta)$.

   $1 + \tan^2(\theta) = \sec^2(\theta)$. Pythagorean identity involving tangent and secant.

2. Find the value of $\tan(\frac{\pi}{4})$.

   $\tan(\frac{\pi}{4}) = 1$. At 45°, tangent equals 1 since opposite equals adjacent.

3. Identify the period of $\tan(\theta)$.

   The period of $\tan(\theta)$ is $\pi$. Tangent completes one cycle every $\pi$ radians.

4. Express $\tan(\frac{\pi}{2} - \theta)$ using a co-function identity.

   $\tan(\frac{\pi}{2} - \theta) = \cot(\theta)$. Co-function identity: tangent of complementary angle equals cotangent.

5. What is the reciprocal identity for secant?

   $\sec(\theta) = \frac{1}{\cos(\theta)}$. Secant is the reciprocal of cosine.

6. Find the value of $\sin(0)$.

   $\sin(0) = 0$. At zero radians, sine equals zero.

7. Which quadrant is $\cos(\theta) > 0$ and $\sin(\theta) < 0$?

   Quadrant IV. In Quadrant IV, cosine is positive and sine is negative.

8. Express cosine in terms of sine for an angle $\theta$.

   $\cos(\theta) = \sin(\frac{\pi}{2} - \theta)$. Co-function identity relating cosine and sine.

9. Convert radians to degrees for $\frac{\pi}{3}$.

   $\frac{\pi}{3}$ radians = $60^\circ$. Multiply by $\frac{180}{\pi}$ to convert radians to degrees.

10. Express $\sin(\theta + \pi)$ using an identity.

    $\sin(\theta + \pi) = -\sin(\theta)$. Adding $\pi$ to angle negates sine value.

11. Convert degrees to radians for $180^\circ$.

    $180^\circ$ = $\pi$ radians. Multiply by $\frac{\pi}{180}$ to convert degrees to radians.

12. Express $\tan(-\theta)$ using an identity.

    $\tan(-\theta) = -\tan(\theta)$. Tangent is an odd function, so $\tan(-x) = -\tan(x)$.

13. Find the value of $\csc(\frac{\pi}{2})$.

    $\csc(\frac{\pi}{2}) = 1$. Since $\sin(\frac{\pi}{2}) = 1$, its reciprocal is 1.

14. Express $\cos(\frac{\pi}{2} - \theta)$ using a co-function identity.

    $\cos(\frac{\pi}{2} - \theta) = \sin(\theta)$. Co-function identity: cosine of complementary angle equals sine.

15. What is the reciprocal identity for cosecant?

    $\csc(\theta) = \frac{1}{\sin(\theta)}$. Cosecant is the reciprocal of sine.

16. Find the value of $\cos(\pi)$.

    $\cos(\pi) = -1$. At $\pi$ radians (180°), cosine equals -1.

17. Express $\cos(\theta + \pi)$ using an identity.

    $\cos(\theta + \pi) = -\cos(\theta)$. Adding $\pi$ to angle negates cosine value.

18. State the identity for $1 + \cot^2(\theta)$.

    $1 + \cot^2(\theta) = \csc^2(\theta)$. Pythagorean identity involving cotangent and cosecant.

19. What is the reciprocal identity for cotangent?

    $\cot(\theta) = \frac{1}{\tan(\theta)}$. Cotangent is the reciprocal of tangent.

20. Convert $\cos(\theta)$ to terms of secant.

    $\cos(\theta) = \frac{1}{\sec(\theta)}$. Reciprocal identity relating cosine and secant.

21. Convert $\sin(\theta)$ to terms of cosecant.

    $\sin(\theta) = \frac{1}{\csc(\theta)}$. Reciprocal identity relating sine and cosecant.

22. Express tangent in terms of sine and cosine for an angle $\theta$.

    $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$. Quotient identity expressing tangent in terms of sine and cosine.

23. Express $\tan(2\theta)$ using a double angle identity.

    $\tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)}$. Double angle identity for tangent function.

24. Find the value of $\sec(0)$.

    $\sec(0) = 1$. Since $\cos(0) = 1$, its reciprocal is 1.

25. What is the sine of an angle expressed as a ratio in a right triangle?

    Sine = $\frac{\text{opposite}}{\text{hypotenuse}}$. Basic right triangle definition for sine function.

26. Express $\cos(-\theta)$ using an identity.

    $\cos(-\theta) = \cos(\theta)$. Cosine is an even function, so $\cos(-x) = \cos(x)$.

27. Express $\sin(\frac{\pi}{2} - \theta)$ using a co-function identity.

    $\sin(\frac{\pi}{2} - \theta) = \cos(\theta)$. Co-function identity: sine of complementary angle equals cosine.

28. Find the value of $\cot(\frac{\pi}{4})$.

    $\cot(\frac{\pi}{4}) = 1$. Since $\tan(\frac{\pi}{4}) = 1$, its reciprocal is 1.

29. Identify the period of $\sin(\theta)$.

    The period of $\sin(\theta)$ is $2\pi$. Sine completes one cycle every $2\pi$ radians.

30. What is the cosine of an angle expressed as a ratio in a right triangle?

    Cosine = $\frac{\text{adjacent}}{\text{hypotenuse}}$. Basic right triangle definition for cosine function.

31. Express sine in terms of cosine for an angle $\theta$.

    $\sin(\theta) = \cos(\frac{\pi}{2} - \theta)$. Co-function identity relating sine and cosine.

32. Which quadrant is $\sin(\theta) > 0$ and $\cos(\theta) < 0$?

    Quadrant II. In Quadrant II, sine is positive and cosine is negative.

33. Express $\cos(2\theta)$ using a double angle identity.

    $\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)$. Double angle identity for cosine function.

34. Convert $\tan(\theta)$ to terms of cotangent.

    $\tan(\theta) = \frac{1}{\cot(\theta)}$. Reciprocal identity relating tangent and cotangent.

35. Express $\sin(2\theta)$ using a double angle identity.

    $\sin(2\theta) = 2\sin(\theta)\cos(\theta)$. Double angle identity for sine function.

36. What is the identity for $\sin^2(\theta) + \cos^2(\theta)$?

    $\sin^2(\theta) + \cos^2(\theta) = 1$. Pythagorean identity, fundamental trigonometric relationship.

37. What is the tangent of an angle expressed as a ratio in a right triangle?

    Tangent = $\frac{\text{opposite}}{\text{adjacent}}$. Basic right triangle definition for tangent function.

38. Express $\sin(-\theta)$ using an identity.

    $\sin(-\theta) = -\sin(\theta)$. Sine is an odd function, so $\sin(-x) = -\sin(x)$.

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