AP Precalculus Flashcards: Trigonometry And Polar Coordinates

What this deck covers

This deck focuses on Trigonometry And Polar Coordinates, giving you a quick way to review the definitions, rules, and examples that matter most for AP 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: State the angle addition formula for cosine.

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

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

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

Flashcard 3: What is the double angle formula for tangent?

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

Flashcard 4: State the Pythagorean identity.

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

Flashcard 5: What is the half-angle formula for tangent?

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

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

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

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

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

Flashcard 8: What are the polar coordinates of the origin?

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

Flashcard 9: What is the double angle formula for tangent?

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

Flashcard 10: What is the half-angle formula for sine?

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

Flashcard 11: What is the half-angle formula for cosine?

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

Flashcard 12: What is the polar form of the complex number $1 + i$?

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

Flashcard 13: What is the cotangent of an angle in terms of tangent?

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

Flashcard 14: State the Pythagorean identity.

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

Flashcard 15: State the formula for converting Cartesian to polar coordinates.

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

Flashcard 16: Convert $(1, \frac{\pi}{3})$ from polar to Cartesian coordinates.

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

Flashcard 17: What is the sine of an angle in a right triangle?

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

Flashcard 18: Find the exact value of $\cos(60^\circ)$.

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

Flashcard 19: What is the half-angle formula for tangent?

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

Flashcard 20: What is the cosecant of an angle in terms of sine?

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

Flashcard 21: Find the exact value of $\sin(30^\circ)$.

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

Flashcard 22: Convert $(1, \frac{\pi}{3})$ from polar to Cartesian coordinates.

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

Flashcard 23: What is the double angle formula for sine?

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

Flashcard 24: What is the law of cosines?

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

Flashcard 25: What is the double angle formula for cosine?

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

Flashcard 26: What is the polar equation of a circle centered at the origin?

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

Flashcard 27: State the angle addition formula for sine.

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

Flashcard 28: What is the cosecant of an angle in terms of sine?

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

Flashcard 29: State the angle addition formula for tangent.

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

Flashcard 30: What is the tangent of an angle in a right triangle?

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

Flashcard 31: Find the exact value of $\sin(30^\circ)$.

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

Flashcard 32: What is the law of sines?

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

Flashcard 33: State the formula for converting polar to Cartesian coordinates.

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

Flashcard 34: What is the polar equation of a line through the origin?

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

Flashcard 35: State the angle addition formula for tangent.

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

Flashcard 36: Convert $\frac{\pi}{4}$ radians to degrees.

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

Flashcard 37: What is the polar form of the complex number $-2 - 2i$?

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

Flashcard 38: What is the polar form of the complex number $-1 + i\sqrt{3}$?

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

Flashcard 39: State the formula for arc length.

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

Flashcard 40: Convert $180^\circ$ to radians.

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

Flashcard 41: What is the double angle formula for cosine?

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

Flashcard 42: What is the tangent of an angle in a right triangle?

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

Flashcard 43: What is the cosine of an angle in a right triangle?

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

Flashcard 44: What is the half-angle formula for sine?

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

Flashcard 45: State the formula for arc length.

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

Flashcard 46: State the formula for converting polar to Cartesian coordinates.

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

Flashcard 47: State the formula for sector area.

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

Flashcard 48: What is the law of cosines?

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

Flashcard 49: State the formula for sector area.

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

Flashcard 50: What is the secant of an angle in terms of cosine?

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

Flashcard 51: What is the polar form of the complex number $1 + i$?

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

Flashcard 52: What is the cosine of an angle in a right triangle?

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

Flashcard 53: Find the reference angle for $210^\circ$.

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

Flashcard 54: Convert $180^\circ$ to radians.

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

Flashcard 55: What are the polar coordinates of the origin?

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

Flashcard 56: What is the law of sines?

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

Flashcard 57: What is the polar equation of a line through the origin?

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

Flashcard 58: Find the exact value of $\cos(60^\circ)$.

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

Flashcard 59: What is the double angle formula for sine?

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

Flashcard 60: Find the exact value of $\tan(45^\circ)$.

Answer:

1. Standard unit circle value.

Flashcard 61: Convert $\frac{\pi}{4}$ radians to degrees.

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

Flashcard 62: State the formula for converting Cartesian to polar coordinates.

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

Flashcard 63: Find the reference angle for $210^\circ$.

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

Flashcard 64: What is the sine of an angle in a right triangle?

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

Flashcard 65: What is the half-angle formula for cosine?

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

Flashcard 66: State the angle addition formula for cosine.

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

Flashcard 67: Find the exact value of $\tan(45^\circ)$.

Answer:

1. Standard unit circle value.

Flashcard 68: What is the polar equation of a circle centered at the origin?

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

Flashcard 69: State the angle addition formula for sine.

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

Flashcard 70: What is the cotangent of an angle in terms of tangent?

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