Unit Circle and Trigonometry Flashcards
Free unit circle and trigonometry flashcards covering exact values for the special angles, degree and radian conversion, SOH-CAH-TOA, the main identities, the law of sines and the law of cosines. Flip, mark and drill.
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Right triangle trig
| Question | Answer |
|---|---|
| What does SOH-CAH-TOA stand for? | sin = opposite / hypotenuse, cos = adjacent / hypotenuse, tan = opposite / adjacent. |
| csc θ equals? | 1 / sin θ |
| sec θ equals? | 1 / cos θ |
| cot θ equals? | 1 / tan θ = cos θ / sin θ |
| tan θ in terms of sin and cos? | tan θ = sin θ / cos θ |
| If sin θ = 3/5 in a right triangle, what is cos θ? | 4/5 (a 3-4-5 triangle) |
| How do you find an angle when you know two sides? | Use an inverse function such as sin⁻¹, cos⁻¹ or tan⁻¹. |
Degrees and radians
| Question | Answer |
|---|---|
| Convert degrees to radians? | Multiply by π/180. |
| Convert radians to degrees? | Multiply by 180/π. |
| 180° in radians? | π |
| 90° in radians? | π/2 |
| 60° in radians? | π/3 |
| 45° in radians? | π/4 |
| 30° in radians? | π/6 |
| 360° in radians? | 2π |
| π/6 in degrees? | 30° |
| 2π/3 in degrees? | 120° |
| 5π/4 in degrees? | 225° |
| 7π/6 in degrees? | 210° |
Special angle values
| Question | Answer |
|---|---|
| sin 0° and cos 0°? | sin 0° = 0, cos 0° = 1 |
| sin 30° and cos 30°? | sin 30° = 1/2, cos 30° = √3/2 |
| sin 45° and cos 45°? | Both √2/2 |
| sin 60° and cos 60°? | sin 60° = √3/2, cos 60° = 1/2 |
| sin 90° and cos 90°? | sin 90° = 1, cos 90° = 0 |
| tan 30°? | √3/3 (1/√3) |
| tan 45°? | 1 |
| tan 60°? | √3 |
| tan 90°? | Undefined |
| sin(π/6), cos(π/6)? | 1/2 and √3/2 |
| sin(π/4), cos(π/4)? | √2/2 and √2/2 |
| sin(π/3), cos(π/3)? | √3/2 and 1/2 |
Unit circle quadrants
| Question | Answer |
|---|---|
| On the unit circle, what are the coordinates of the point at angle θ? | (cos θ, sin θ) |
| What is the equation of the unit circle? | x² + y² = 1 |
| Which trig functions are positive in Quadrant I? | All of them (All Students Take Calculus: All, Sine, Tangent, Cosine) |
| Which are positive in Quadrant II? | Sine (and cosecant) |
| Which are positive in Quadrant III? | Tangent (and cotangent) |
| Which are positive in Quadrant IV? | Cosine (and secant) |
| Point on the unit circle at 120°? | (−1/2, √3/2) |
| Point on the unit circle at 135°? | (−√2/2, √2/2) |
| Point on the unit circle at 210°? | (−√3/2, −1/2) |
| Point on the unit circle at 270°? | (0, −1) |
| Point on the unit circle at 315°? | (√2/2, −√2/2) |
| sin 150° and cos 150°? | 1/2 and −√3/2 |
| sin 225° and cos 225°? | Both −√2/2 |
| sin 300° and cos 300°? | −√3/2 and 1/2 |
Identities
| Question | Answer |
|---|---|
| Pythagorean identity? | sin²θ + cos²θ = 1 |
| Second Pythagorean identity? | 1 + tan²θ = sec²θ |
| Third Pythagorean identity? | 1 + cot²θ = csc²θ |
| sin(−θ)? | −sin θ (sine is odd) |
| cos(−θ)? | cos θ (cosine is even) |
| tan(−θ)? | −tan θ |
| sin(90° − θ)? | cos θ |
| cos(90° − θ)? | sin θ |
| Double angle: sin 2θ? | 2 sin θ cos θ |
| Double angle: cos 2θ? | cos²θ − sin²θ = 1 − 2sin²θ = 2cos²θ − 1 |
| Double angle: tan 2θ? | 2 tan θ / (1 − tan²θ) |
| sin(A + B)? | sin A cos B + cos A sin B |
| cos(A + B)? | cos A cos B − sin A sin B |
| tan(A + B)? | (tan A + tan B) / (1 − tan A tan B) |
Triangle laws and graphs
| Question | Answer |
|---|---|
| Law of sines? | a / sin A = b / sin B = c / sin C |
| Law of cosines? | c² = a² + b² − 2ab cos C |
| Area of a triangle from two sides and the included angle? | A = ½ ab sin C |
| Period of y = sin x and y = cos x? | 2π (360°) |
| Period of y = tan x? | π (180°) |
| Amplitude of y = A sin(Bx)? | |A| |
| Period of y = A sin(Bx)? | 2π / |B| |
| Range of sin x and cos x? | [−1, 1] |
| When does sin x = 0? | At x = 0, π, 2π and every multiple of π |
Frequently asked questions
What is the easiest way to memorize the unit circle?
Learn the first quadrant (30°, 45°, 60°) with their √ values, then use reference angles and the signs by quadrant (All Students Take Calculus) for the other three quadrants.
Do I need to know radians for the SAT or ACT?
Radian measure appears on both tests, especially in circle and trigonometry questions, so it is worth knowing the common conversions.
What is a reference angle?
The acute angle between the terminal side of an angle and the x-axis. Trig values of an angle equal those of its reference angle, up to sign.
Sources
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