ACT Math: Trigonometry
Trigonometry on the ACT goes a step beyond just SOH-CAH-TOA, roughly half of ACT trig questions draw on the Law of Sines and Cosines, radians and the unit circle, trig identities, and the graphs of sine and cosine. This free, complete lesson covers SOH-CAH-TOA and the Pythagorean identity, special angle values, radians and the unit circle, the Law of Sines and Law of Cosines, and graphs of sine and cosine (amplitude and period). Each idea is followed by a fully worked example, and then a set of original practice problems with instant feedback, in the same four-answer-choice, calculator-allowed format as the current ACT. Everything here is free, and you can keep practicing afterward with the full ACT Math Question Bank linked below.
1. SOH-CAH-TOA and the Pythagorean Identity
sin θ = oppositehypotenuse cos θ = adjacenthypotenuse tan θ = oppositeadjacent
For any angle θ, the Pythagorean identity holds:
sin² θ + cos² θ = 1
Worked Example: In a right triangle, angle θ has opposite side 6, adjacent side 8, hypotenuse 10. Find sin θ, cos θ, and tan θ.
sin θ = 610 = 35 cos θ = 810 = 45 tan θ = 68 = 34
1. In a right triangle, angle θ has an opposite side of 9 and a hypotenuse of 15. Find sin θ.
Explanation: sin θ = 915 = 35.
2. In a right triangle, angle θ has an adjacent side of 12 and a hypotenuse of 13. Find cos θ.
Explanation: cos θ = 1213.
3. In a right triangle, angle θ has an opposite side of 7 and an adjacent side of 24. Find tan θ.
Explanation: tan θ = 724.
4. If sin θ = 35, and θ is acute, find cos θ using the Pythagorean identity.
Explanation: 925 + cos² θ = 1, so cos² θ = 1625, giving cos θ = 45.
5. If cos θ = 513, and θ is acute, find sin θ using the Pythagorean identity.
Explanation: sin² θ + 25169 = 1, so sin² θ = 144169, giving sin θ = 1213.
2. Special Angle Values
The sine, cosine, and tangent of 30°, 45°, and 60° come directly from the special right triangles, and are worth memorizing outright.
| θ | sin θ | cos θ | tan θ |
|---|---|---|---|
| 30° | 12 | √32 | √33 |
| 45° | √22 | √22 | 1 |
| 60° | √32 | 12 | √3 |
Worked Example: Find sin(30°), cos(30°), and tan(30°) using the 30-60-90 triangle (side ratio 1 : √3 : 2).
For the 30° angle: opposite = 1, adjacent = √3, hypotenuse = 2.
sin(30°) = 12 cos(30°) = √32 tan(30°) = 1√3 = √33
1. What is sin(45°)?
Explanation: A 45-45-90 triangle has legs 1 and hypotenuse √2: sin(45°) = 1√2 = √22.
2. What is cos(60°)?
Explanation: For the 60° angle in a 30-60-90 triangle, the adjacent side is 1 and the hypotenuse is 2: cos(60°) = 12.
3. What is tan(45°)?
Explanation: Since the two legs of a 45-45-90 triangle are equal, tan(45°) = oppositeadjacent = 1.
4. What is sin(90°)?
Explanation: sin(90°) = 1, the maximum possible value of sine.
5. What is cos(0°)?
Explanation: cos(0°) = 1, the maximum possible value of cosine.
3. Radians and the Unit Circle
A full circle is 360°, which equals 2π radians. The unit circle, a circle of radius 1 centered at the origin, defines sine and cosine for any angle as the y- and x-coordinates of the point where that angle meets the circle.
radians = degrees × π180 degrees = radians × 180π
Worked Example: Convert 120° to radians.
120 × π180 = 2π3
1. Convert 270° to radians.
Explanation: 270 × π180 = 3π2.
2. Convert 5π6 radians to degrees.
Explanation: 5π6 × 180π = 150°.
3. On the unit circle, what is the value of cos(0)?
Explanation: An angle of 0 corresponds to the point (1, 0) on the unit circle, so cos(0) = 1 (the x-coordinate).
4. On the unit circle, what is the value of sin(π2)?
Explanation: An angle of π2 (90°) corresponds to the point (0, 1) on the unit circle, so sin(π2) = 1 (the y-coordinate).
5. In which quadrant is an angle of 5π4 radians located?
Explanation: 5π4 radians equals 225°, which falls between 180° and 270°, placing it in Quadrant III.
4. Law of Sines and Law of Cosines
For any triangle (not just right triangles) with sides a, b, c opposite angles A, B, C:
Law of Sines: asin A = bsin B = csin C
Law of Cosines: a² = b² + c² − 2bc·cos A
1. Which of the following correctly represents the Law of Sines for triangle ABC, with sides a, b, c opposite angles A, B, C respectively?
Explanation: The Law of Sines pairs each side with the sine of its own opposite angle: asin A = bsin B = csin C.
2. Which of the following correctly represents the Law of Cosines for finding side a, opposite angle A?
Explanation: The Law of Cosines is a² = b² + c² − 2bc·cos A, note the minus sign before the last term.
3. In a triangle, angle A = 30°, side a = 5, and angle B = 90°. Using the Law of Sines, what is the length of side b?
Explanation: 5sin 30° = bsin 90°. Since sin 30° = 0.5 and sin 90° = 1: 50.5 = 10 = b1, so b = 10.
4. In triangle ABC, sides b = 7, c = 9, and angle A = 60°. Using the Law of Cosines, what is a² (before taking the square root)?
Explanation: a² = 7² + 9² − 2(7)(9)cos(60°) = 49 + 81 − 126(0.5) = 130 − 63 = 67.
5. Graphs of Sine and Cosine
The graphs of y = sin(x) and y = cos(x) are smooth, repeating waves. For y = A sin(Bx), the amplitude is |A| (how far the wave reaches above and below the midline), and the period (the length of one complete wave) is 2π|B|.
Worked Example: For y = 3 sin(x), what is the amplitude?
amplitude = |3| = 3
1. For y = 4 cos(x), what is the amplitude?
Explanation: The amplitude is |4| = 4.
2. What is the period of y = sin(x)?
Explanation: The standard sine function completes one full wave every 2π.
3. For y = sin(2x), what is the period?
Explanation: period = 2π|B| = 2π2 = π.
4. For y = 5 sin(x), what are the maximum and minimum values of y?
Explanation: The amplitude of 5 means the graph reaches 5 units above and below its midline (0), giving a maximum of 5 and a minimum of −5.
Common Mistakes to Avoid
- Mixing up which side is opposite versus adjacent. These labels are always relative to the specific angle in question, not fixed positions in the triangle.
- Using SOH-CAH-TOA on a non-right triangle. The basic trig ratios only work in right triangles. For any other triangle, switch to the Law of Sines or Law of Cosines.
- Forgetting the minus sign in the Law of Cosines. It's a² = b² + c² − 2bc·cos A, not plus, this is easy to misremember since it resembles the Pythagorean theorem.
- Leaving the calculator in radian mode for a degree-based problem, or vice versa. Always confirm your calculator's angle mode matches the units given in the problem before computing.
- Confusing amplitude with period. Amplitude describes the wave's height (vertical stretch); period describes how long it takes to repeat (horizontal length). They come from different parts of the equation.
Frequently Asked Questions
Is it really true that ACT trigonometry goes beyond SOH-CAH-TOA?
Yes. While right-triangle ratios cover roughly half of ACT trig questions, a well-rounded prep plan should also include the Law of Sines and Cosines, radian measure and the unit circle, and the basic shape and parameters of sine and cosine graphs.
When should I use the Law of Sines versus the Law of Cosines?
Use the Law of Sines when you know an angle and its opposite side, plus one more piece of information (another angle, or another side-angle pair). Use the Law of Cosines when you know two sides and the included angle, or all three sides, situations where the Law of Sines doesn't have enough matching angle-side pairs to work with.
Do I need to memorize every value on the unit circle?
Knowing the special angles, 0°, 30°, 45°, 60°, and 90°, along with which quadrant an angle falls in, covers the large majority of what's tested. Full unit circle memorization for every angle is not necessary.
Where can I practice more problems like these?
The Trigonometry quizzes in the ACT Math Question Bank include additional original problems on this topic, along with quizzes covering every other topic tested on the ACT Math section.