ACT Math Flashcards: Trigonometry

What this deck covers

This deck focuses on Trigonometry, giving you a quick way to review the definitions, rules, and examples that matter most for ACT Math.

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: What is $\cos\left(\frac{\pi}{3}\right)$?

Answer: $\frac{1}{2}$. From 60° angle in 30-60-90 triangle.

Flashcard 2: What is the angle of depression if the adjacent side is 1 and the opposite side is 1?

Answer: 45 degrees. Equal opposite and adjacent sides create a 45° angle.

Flashcard 3: What is the secant of a 60-degree angle?

Answer: $2$. Secant is $\frac{1}{\cos(60^\circ)} = \frac{1}{1/2}$

Flashcard 4: What is the arc length formula for a circle with radius $r$ and central angle $\theta$ (in radians)?

Answer: $s=r\theta$. Arc length equals radius times central angle in radians.

Flashcard 5: What is the cotangent of an angle in terms of tangent?

Answer: $\text{cot} \theta = \frac{1}{\text{tan} \theta}$. Cotangent is the reciprocal of tangent.

Flashcard 6: What is $\sin(2\pi)$ on the unit circle?

Answer: $0$. At $2\pi$, back to (1,0), so sine is 0.

Flashcard 7: What is the value of $\text{cos}(90^\text{o})$?

Answer:

1. At 90°, there's no horizontal component.

Flashcard 8: What are the side ratios for a $45^\circ$-$45^\circ$-$90^\circ$ triangle (legs to hypotenuse)?

Answer: $1:1:\sqrt{2}$. Isosceles right triangle with legs of 1 and hypotenuse $\sqrt{2}$.

Flashcard 9: What is the period of $y=\tan(Bx)$ in terms of $B$?

Answer: $\frac{\pi}{|B|}$. Tangent has period $\pi$, so with coefficient $B$ it's $\frac{\pi}{|B|}$.

Flashcard 10: What is the cosine of a 0-degree angle?

Answer:

1. At 0°, the point is at (1,0) on the unit circle.

Flashcard 11: Find the value of $\text{tan}^{-1}(1)$ in degrees.

Answer: $45^\text{o}$. Inverse tangent of 1 gives the angle where $\tan(\theta) = 1$.

Flashcard 12: What is the degree-to-radian conversion formula for an angle $\theta$ in degrees?

Answer: $\theta_{\text{rad}}=\theta_{\circ}\cdot\frac{\pi}{180}$. Multiply degrees by $\frac{\pi}{180}$ to convert to radians.

Flashcard 13: What is $\sin(2\pi)$ on the unit circle?

Answer: $0$. At $2\pi$, back to (1,0), so sine is 0.

Flashcard 14: What is $\cos\left(\frac{\pi}{2}\right)$ on the unit circle?

Answer: $0$. At $\frac{\pi}{2}$, the point is at (0,1), so cosine is 0.

Flashcard 15: What is the sine of an angle whose secant is 2?

Answer: $\frac{\sqrt{3}}{2}$. If $\sec = 2$, then $\cos = \frac{1}{2}$, use Pythagorean identity.

Flashcard 16: State the double angle formula for sine.

Answer: $\text{sin}(2\theta) = 2\text{sin}(\theta)\text{cos}(\theta)$. Double angle formula using product-to-sum identity.

Flashcard 17: What is the cosecant of a 30-degree angle?

Answer: $2$. Cosecant is $\frac{1}{\sin(30°)} = \frac{1}{1/2}$.

Flashcard 18: What is the secant of a 90-degree angle?

Answer: Undefined. Division by zero makes secant undefined.

Flashcard 19: What is the sine of a $30^\text{o}$ angle?

Answer: $\frac{1}{2}$. This is a special angle value from the 30-60-90 triangle.

Flashcard 20: What is the range of the cosine function?

Answer: [-1, 1]. Cosine oscillates between -1 and 1 for all angles.

Flashcard 21: What is the radian-to-degree conversion formula for an angle $\theta$ in radians?

Answer: $\theta_{\circ}=\theta_{\text{rad}}\cdot\frac{180}{\pi}$. Multiply radians by $\frac{180}{\pi}$ to convert to degrees.

Flashcard 22: What is the secant of an angle in terms of cosine?

Answer: $\text{sec} \theta = \frac{1}{\text{cos} \theta}$. Secant is the reciprocal of cosine.

Flashcard 23: What is the reciprocal of tangent?

Answer: Cotangent. Reciprocal function: $\cot(\theta) = \frac{1}{\tan(\theta)}$.

Flashcard 24: What is $\sin(0)$ on the unit circle?

Answer: $0$. At angle 0, the point is at (1,0), so sine is 0.

Flashcard 25: What is the arc length formula for a circle with radius $r$ and central angle $\theta$ (in radians)?

Answer: $s=r\theta$. Arc length equals radius times central angle in radians.

Flashcard 26: What is the cosecant of an angle in terms of sine?

Answer: $\text{csc} \theta = \frac{1}{\text{sin} \theta}$. Cosecant is the reciprocal of sine.

Flashcard 27: What is the cosine of a $90^\text{o}$ angle?

Answer:

1. At 90°, the adjacent side is zero.

Flashcard 28: What is the value of $\text{tan}(0^\text{o})$?

Answer:

1. At 0°, opposite side has zero length.

Flashcard 29: State the Pythagorean identity for sine and cosine.

Answer: $\text{sin}^2 \theta + \text{cos}^2 \theta = 1$. Fundamental identity from the unit circle definition.

Flashcard 30: What is the radian-to-degree conversion formula for an angle $\theta$ in radians?

Answer: $\theta_{\circ}=\theta_{\text{rad}}\cdot\frac{180}{\pi}$. Multiply radians by $\frac{180}{\pi}$ to convert to degrees.

Flashcard 31: What is $\tan\left(\frac{\pi}{6}\right)$?

Answer: $\frac{\sqrt{3}}{3}$. $\frac{\sin(\pi/6)}{\cos(\pi/6)} = \frac{1/2}{\sqrt{3}/2} = \frac{1}{\sqrt{3}}$.

Flashcard 32: What is the cosecant of a 30-degree angle?

Answer: $2$. Cosecant is $\frac{1}{\sin(30°)} = \frac{1}{1/2}$.

Flashcard 33: What is the amplitude of $y=A\sin(Bx)$ or $y=A\cos(Bx)$ in terms of $A$?

Answer: $|A|$. Maximum distance from equilibrium position is $|A|$.

Flashcard 34: Find the angle if the tangent is 1. Provide the acute angle.

Answer: 45 degrees. Use inverse tangent to find the reference angle.

Flashcard 35: What is the sector area formula for radius $r$ and central angle $\theta$ (in radians)?

Answer: $A=\frac{1}{2}r^2\theta$. Sector area is half the product of radius squared and angle.

Flashcard 36: What is the angle of elevation if the opposite side is 1 and the adjacent side is $\sqrt{3}$?

Answer: 30 degrees. Use $\tan^{-1}(\frac{1}{\sqrt{3}})$ to find the angle.

Flashcard 37: Convert $\frac{\text{π}}{2}$ radians to degrees.

Answer: 90 degrees. $\frac{\pi}{2}$ radians equals 90° by definition.

Flashcard 38: What is the period of $y = \cos(2x)$?

Answer: $\pi$. Period formula: $\frac{2\pi}{|b|}$ where $b = 2$.

Flashcard 39: What is the cotangent of a 45-degree angle?

Answer:

1. At 45°, sine equals cosine, so their ratio is 1.

Flashcard 40: State the Pythagorean identity for cotangent and cosecant.

Answer: $1 + \text{cot}^2 \theta = \text{csc}^2 \theta$. Derived from the Pythagorean identity and reciprocals.

Flashcard 41: What is the cosine of an angle whose cosecant is 2?

Answer: $\frac{\sqrt{3}}{2}$. If $\csc = 2$, then $\sin = \frac{1}{2}$, use Pythagorean identity.

Flashcard 42: What is $\sin\left(\frac{\pi}{4}\right)$?

Answer: $\frac{\sqrt{2}}{2}$. From 45° angle in 45-45-90 triangle.

Flashcard 43: What is the cosine of a 60-degree angle?

Answer: $\frac{1}{2}$. 60° is a special angle in the unit circle.

Flashcard 44: What is the tangent of a 45-degree angle?

Answer:

1. 45° creates an isosceles right triangle.

Flashcard 45: Convert $\frac{\text{π}}{3}$ radians to degrees.

Answer: 60 degrees. Multiply radians by $\frac{180}{\pi}$.

Flashcard 46: State the Pythagorean identity for tangent and secant.

Answer: $1 + \text{tan}^2 \theta = \text{sec}^2 \theta$. Derived from the Pythagorean identity and reciprocals.

Flashcard 47: What is the tangent of $0^\text{o}$?

Answer:

1. Since $\sin(0°) = 0$ and $\cos(0°) = 1$.

Flashcard 48: Find the angle if the sine is $\frac{1}{2}$. Provide the acute angle.

Answer: 30 degrees. Use inverse sine to find the reference angle.

Flashcard 49: Find the angle if the sine is $\frac{1}{2}$. Provide the acute angle.

Answer: 30 degrees. Use inverse sine to find the reference angle.

Flashcard 50: State the formula for the tangent of an angle.

Answer: $\text{tan} \theta = \frac{\text{sin} \theta}{\text{cos} \theta}$. Tangent equals the ratio of sine to cosine.

Flashcard 51: What is the degree-to-radian conversion formula for an angle $\theta$ in degrees?

Answer: $\theta_{\text{rad}}=\theta_{\circ}\cdot\frac{\pi}{180}$. Multiply degrees by $\frac{\pi}{180}$ to convert to radians.

Flashcard 52: What is $\sin(0)$ on the unit circle?

Answer: $0$. At angle 0, the point is at (1,0), so sine is 0.

Flashcard 53: Find the angle if the cosine is $\frac{1}{2}$. Provide the acute angle.

Answer: 60 degrees. Use inverse cosine to find the reference angle.

Flashcard 54: State the reciprocal identity for tangent.

Answer: $\text{tan} \theta = \frac{1}{\text{cot} \theta}$. Tangent and cotangent are reciprocals.

Flashcard 55: State the co-function identity for sine and cosine.

Answer: $\text{sin}(\theta) = \text{cos}(90^\text{o} - \theta)$. Complementary angles have swapped sine and cosine.

Flashcard 56: What is $\tan\left(\frac{\pi}{3}\right)$?

Answer: $\sqrt{3}$. $\frac{\sin(\pi/3)}{\cos(\pi/3)} = \frac{\sqrt{3}/2}{1/2} = \sqrt{3}$.

Flashcard 57: State the Pythagorean identity for sine and cosine.

Answer: $\text{sin}^2 \theta + \text{cos}^2 \theta = 1$. Fundamental identity from the unit circle definition.

Flashcard 58: What is the tangent of a $45^\text{o}$ angle?

Answer:

1. In a 45-45-90 triangle, opposite and adjacent sides are equal.

Flashcard 59: What is $\cos(2\pi)$ on the unit circle?

Answer: $1$. At $2\pi$, back to (1,0), so cosine is 1.

Flashcard 60: What is the cosine of a $60^\text{o}$ angle?

Answer: $\frac{1}{2}$. From the 30-60-90 triangle, $\cos(60°) = \sin(30°)$.

Flashcard 61: What is the cosine of a 60-degree angle?

Answer: $\frac{1}{2}$. 60° is a special angle in the unit circle.

Flashcard 62: Find the angle if the tangent is 1. Provide the acute angle.

Answer: 45 degrees. Use inverse tangent to find the reference angle.

Flashcard 63: What is the period of $y=\sin(Bx)$ or $y=\cos(Bx)$ in terms of $B$?

Answer: $\frac{2\pi}{|B|}$. Time for one complete cycle is $\frac{2\pi}{|B|}$.

Flashcard 64: What is the sine of a 90-degree angle?

Answer:

1. At 90°, the point is at the top of the unit circle.

Flashcard 65: Convert 180 degrees to radians.

Answer: $\pi$. 180° equals $\pi$ radians by definition.

Flashcard 66: What is the cosine of $0^\text{o}$?

Answer:

1. At 0°, the adjacent side equals the hypotenuse.

Flashcard 67: What is the angle of elevation if the opposite side is 1 and the adjacent side is $\text{√3}$?

Answer: 30 degrees. Use $\tan^{-1}(\frac{1}{\sqrt{3}})$ to find the angle.

Flashcard 68: What is the secant of a 90-degree angle?

Answer: Undefined. Division by zero makes secant undefined.

Flashcard 69: Convert $\frac{\text{π}}{3}$ radians to degrees.

Answer: 60 degrees. Multiply radians by $\frac{180}{\pi}$.

Flashcard 70: For which angle is $\text{sin} \theta = 1$?

Answer: $90^\text{o}$. Sine reaches its maximum value of 1 at 90°.

Flashcard 71: What is the sine of a 30-degree angle?

Answer: $\frac{1}{2}$. 30° is a special angle in the unit circle.

Flashcard 72: What is the value of $\text{sin}(0^\text{o})$?

Answer:

1. At 0°, there's no vertical component.

Flashcard 73: What is the sine of $180^\text{o}$?

Answer:

1. At 180°, we're on the negative x-axis where y = 0.

Flashcard 74: Convert 45 degrees to radians.

Answer: $\frac{\text{π}}{4}$. Multiply degrees by $\frac{\pi}{180}$.

Flashcard 75: What is the sine of $0^\text{o}$?

Answer:

1. At 0°, the opposite side is zero.

Flashcard 76: What is $\sin\left(\frac{\pi}{2}\right)$ on the unit circle?

Answer: $1$. At $\frac{\pi}{2}$, the point is at (0,1), so sine is 1.

Flashcard 77: What is the cosecant of a 0-degree angle?

Answer: Undefined. Division by zero makes cosecant undefined.

Flashcard 78: State the double angle formula for cosine.

Answer: $\text{cos}(2\theta) = \text{cos}^2(\theta) - \text{sin}^2(\theta)$. Double angle formula from cosine addition identity.

Flashcard 79: What are the side ratios for a $45^\circ$-$45^\circ$-$90^\circ$ triangle (legs to hypotenuse)?

Answer: $1:1:\sqrt{2}$. Isosceles right triangle with legs of 1 and hypotenuse $\sqrt{2}$.

Flashcard 80: What is the sector area formula for radius $r$ and central angle $\theta$ (in radians)?

Answer: $A=\frac{1}{2}r^2\theta$. Sector area is half the product of radius squared and angle.

Flashcard 81: What are the side ratios for a $30^\circ$-$60^\circ$-$90^\circ$ triangle (short:long:hyp)?

Answer: $1:\sqrt{3}:2$. Half of an equilateral triangle: short leg, long leg, hypotenuse.

Flashcard 82: What are the right-triangle definitions of $\sin(\theta)$, $\cos(\theta)$, and $\tan(\theta)$?

Answer: $\sin=\frac{\text{opp}}{\text{hyp}},\ \cos=\frac{\text{adj}}{\text{hyp}},\ \tan=\frac{\text{opp}}{\text{adj}}$. SOH-CAH-TOA: ratios of opposite, adjacent sides to hypotenuse.

Flashcard 83: What is the tangent of a 45-degree angle?

Answer:

1. 45° creates an isosceles right triangle.

Flashcard 84: What is the period of $y=\tan(Bx)$ in terms of $B$?

Answer: $\frac{\pi}{|B|}$. Tangent has period $\pi$, so with coefficient $B$ it's $\frac{\pi}{|B|}$.

Flashcard 85: What is the cosecant of a 0-degree angle?

Answer: Undefined. Division by zero makes cosecant undefined.

Flashcard 86: What is $\cos(\pi)$ on the unit circle?

Answer: $-1$. At $\pi$, the point is at (-1,0), so cosine is -1.

Flashcard 87: What is $\tan\left(\frac{\pi}{4}\right)$?

Answer: $1$. $\frac{\sin(\pi/4)}{\cos(\pi/4)} = \frac{\sqrt{2}/2}{\sqrt{2}/2} = 1$.

Flashcard 88: What are the side ratios for a $30^\circ$-$60^\circ$-$90^\circ$ triangle (short:long:hyp)?

Answer: $1:\sqrt{3}:2$. Half of an equilateral triangle: short leg, long leg, hypotenuse.

Flashcard 89: What is the cotangent of a 0-degree angle?

Answer: Undefined. Division by zero makes cotangent undefined.

Flashcard 90: What is the sine of a $90^\text{o}$ angle?

Answer:

1. At 90°, the opposite side equals the hypotenuse.

Flashcard 91: What is $\sin(\pi)$ on the unit circle?

Answer: $0$. At $\pi$, the point is at (-1,0), so sine is 0.

Flashcard 92: What is $\cos(\pi)$ on the unit circle?

Answer: $-1$. At $\pi$, the point is at (-1,0), so cosine is -1.

Flashcard 93: What is the period of $y=\sin(Bx)$ or $y=\cos(Bx)$ in terms of $B$?

Answer: $\frac{2\pi}{|B|}$. Time for one complete cycle is $\frac{2\pi}{|B|}$.

Flashcard 94: What is the tangent of a 90-degree angle?

Answer: Undefined. Division by zero makes tangent undefined.

Flashcard 95: What is the formula for the distance between two points $(x_1,y_1)$ and $(x_2,y_2)$ in the plane?

Answer: $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. Pythagorean theorem applied to coordinate differences.

Flashcard 96: Convert $\frac{\text{π}}{2}$ radians to degrees.

Answer: 90 degrees. $\frac{\pi}{2}$ radians equals 90° by definition.

Flashcard 97: What is the sine of a 30-degree angle?

Answer: $\frac{1}{2}$. 30° is a special angle in the unit circle.

Flashcard 98: What is the tangent of $180^\text{o}$?

Answer:

1. Since $\sin(180°) = 0$ and $\cos(180°) = -1$.

Flashcard 99: What is the cotangent of a 45-degree angle?

Answer:

1. At 45°, sine equals cosine, so their ratio is 1.

Flashcard 100: What is $\tan\left(\frac{\pi}{4}\right)$?

Answer: $1$. $\frac{\sin(\pi/4)}{\cos(\pi/4)} = \frac{\sqrt{2}/2}{\sqrt{2}/2} = 1$.