Trigonometry and Polar Coordinates - AP Precalculus

1. State the angle addition formula for cosine.

   $\cos(a + b) = \cos a \cos b - \sin a \sin b$. Used to find cosine of sum of two angles.

2. What is the secant of an angle in terms of cosine?

   Secant = $\frac{1}{\cos \theta}$. Reciprocal function of cosine.

3. What is the double angle formula for tangent?

   $\tan(2\theta) = \frac{2\tan \theta}{1 - \tan^2 \theta}$. Derived from angle addition formulas.

4. State the Pythagorean identity.

   $\sin^2 \theta + \cos^2 \theta = 1$. Fundamental trigonometric identity from unit circle.

5. What is the half-angle formula for tangent?

   $\tan(\frac{\theta}{2}) = \pm \sqrt{\frac{1 - \cos \theta}{1 + \cos \theta}}$. Sign depends on quadrant of $\frac{\theta}{2}$.

6. What is the polar form of the complex number $-1 + i\sqrt{3}$?

   $2(\cos \frac{2\pi}{3} + i \sin \frac{2\pi}{3})$. Magnitude $2$, argument $\frac{2\pi}{3}$.

7. What is the polar form of the complex number $-2 - 2i$?

   $2\sqrt{2}(\cos \frac{5\pi}{4} + i \sin \frac{5\pi}{4})$. Magnitude $2\sqrt{2}$, argument $\frac{5\pi}{4}$.

8. What are the polar coordinates of the origin?

   $(0, 0)$. The pole has radius $0$ and undefined angle.

9. What is the polar form of the complex number $1 + i$?

   $\sqrt{2}(\cos \frac{\pi}{4} + i \sin \frac{\pi}{4})$. Magnitude $\sqrt{2}$, argument $\frac{\pi}{4}$.

10. What is the sine of an angle in a right triangle?

    Sine = Opposite side / Hypotenuse. Basic trigonometric ratio in right triangles.

11. Find the exact value of $\cos(60^\circ)$.

    $\frac{1}{2}$. Standard unit circle value.

12. Convert $(1, \frac{\pi}{3})$ from polar to Cartesian coordinates.

    $(0.5, 0.866)$. Use $x = r\cos\theta$, $y = r\sin\theta$.

13. What is the double angle formula for sine?

    $\sin(2\theta) = 2\sin \theta \cos \theta$. Derived from angle addition formulas.

14. What is the law of cosines?

    $c^2 = a^2 + b^2 - 2ab\cos C$. Relates all three sides and one angle.

15. State the angle addition formula for sine.

    $\sin(a + b) = \sin a \cos b + \cos a \sin b$. Used to find sine of sum of two angles.

16. What is the cosecant of an angle in terms of sine?

    Cosecant = $\frac{1}{\sin \theta}$. Reciprocal function of sine.

17. Find the exact value of $\sin(30^\circ)$.

    $\frac{1}{2}$. Standard unit circle value.

18. What is the polar equation of a line through the origin?

    $\theta = \alpha$. All points on ray from origin at angle $\alpha$.

19. State the angle addition formula for tangent.

    $\tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b}$. Used to find tangent of sum of two angles.

20. Convert $\frac{\pi}{4}$ radians to degrees.

    $45^\circ$. Use $\frac{\pi}{4} \times \frac{180^\circ}{\pi}$.

21. Convert $180^\circ$ to radians.

    $\pi$ radians. Use $180^\circ \times \frac{\pi}{180^\circ}$.

22. What is the double angle formula for cosine?

    $\cos(2\theta) = \cos^2 \theta - \sin^2 \theta$. Derived from angle addition formulas.

23. What is the tangent of an angle in a right triangle?

    Tangent = Opposite side / Adjacent side. Basic trigonometric ratio in right triangles.

24. What is the cosine of an angle in a right triangle?

    Cosine = Adjacent side / Hypotenuse. Basic trigonometric ratio in right triangles.

25. What is the half-angle formula for sine?

    $\sin(\frac{\theta}{2}) = \pm \sqrt{\frac{1 - \cos \theta}{2}}$. Sign depends on quadrant of $\frac{\theta}{2}$.

26. State the formula for arc length.

    Arc Length = $r \theta$. Where $\theta$ is in radians.

27. State the formula for converting polar to Cartesian coordinates.

    $x = r \cos \theta, y = r \sin \theta$. Use trigonometric definitions.

28. State the formula for sector area.

    Sector Area = $\frac{1}{2} r^2 \theta$. Where $\theta$ is in radians.

29. Find the reference angle for $210^\circ$.

    $30^\circ$. Subtract $180^\circ$ from third quadrant angle.

30. What is the law of sines?

    $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$. Ratios of sides to opposite angles are equal.

31. State the formula for converting Cartesian to polar coordinates.

    $r = \sqrt{x^2 + y^2}, \theta = \tan^{-1}(\frac{y}{x})$. Distance formula and arctangent function.

32. What is the half-angle formula for cosine?

    $\cos(\frac{\theta}{2}) = \pm \sqrt{\frac{1 + \cos \theta}{2}}$. Sign depends on quadrant of $\frac{\theta}{2}$.

33. Find the exact value of $\tan(45^\circ)$.

    1. Standard unit circle value.

34. What is the polar equation of a circle centered at the origin?

    $r = a$. All points equidistant from origin.

35. What is the cotangent of an angle in terms of tangent?

    Cotangent = $\frac{1}{\tan \theta}$. Reciprocal function of tangent.

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