Right Triangle Trigonometry SOHCAHTOA — ACT Math Guide
Right triangle trigonometry SOHCAHTOA ACT problems appear consistently on test day and require you to find missing sides or angles using sine, cosine, and tangent ratios. These problems involve applying the famous SOHCAHTOA memory device to solve real-world scenarios like finding heights of buildings or distances across rivers. Trigonometry questions make up about 4-6 problems out of the 60 questions in 60 minutes on the ACT math section, so mastering this topic can significantly boost your ACT math score. With the right approach and practice, you'll tackle these problems with confidence.
What You Need to Know
SOHCAHTOA stands for Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent
The hypotenuse is always the longest side, opposite the right angle
Opposite side is across from the angle you're working with
Adjacent side touches the angle you're working with (but isn't the hypotenuse)
Use inverse trig functions (sin⁻¹, cos⁻¹, tan⁻¹) to find angles when given sides
Your calculator is allowed throughout the entire ACT math section — use it for trig calculations
📐 KEY FORMULA: SOH: sin θ = opposite/hypotenuse, CAH: cos θ = adjacent/hypotenuse, TOA: tan θ = opposite/adjacent
⏱️ ACT TIME TIP: Draw and label your triangle immediately — this visual saves time and prevents errors when working through 60 questions in 60 minutes.
How to Solve Right Triangle Trigonometry SOHCAHTOA on the ACT
Example Question 1 — Easy/Medium Difficulty
A ladder leans against a wall, making a 65° angle with the ground. If the ladder is 12 feet long, how high up the wall does the ladder reach?
A) 10.9 feet
B) 11.2 feet
C) 5.1 feet
D) 12.0 feet
E) 13.2 feet
Solution:
Step 1: Draw a right triangle with the ladder as the hypotenuse (12 feet), the wall as the opposite side, and the ground as the adjacent side.
Step 2: Identify that you need the opposite side and have the hypotenuse, so use SOH: sin 65° = opposite/12.
Step 3: Solve for the opposite: opposite = 12 × sin 65° = 12 × 0.906 = 10.9 feet.
✅Answer: A — The ladder reaches 10.9 feet up the wall using sine since we had the hypotenuse and needed the opposite side.
Example Question 2 — Hard Difficulty
From the top of a 150-foot lighthouse, the angle of depression to a boat is 18°. What is the horizontal distance from the base of the lighthouse to the boat?
A) 461 feet
B) 487 feet
C) 158 feet
D) 48 feet
E) 142 feet
Solution:
Step 1: Draw a right triangle where the lighthouse height (150 feet) is the opposite side to the 18° angle, and the horizontal distance is the adjacent side.
Step 2: Use TOA since you have opposite and need adjacent: tan 18° = 150/adjacent.
Step 3: Solve: adjacent = 150/tan 18° = 150/0.325 = 462 feet ≈ 461 feet.
✅Answer: A — We used tangent because we had the opposite side and needed the adjacent side.
Common ACT Math Mistakes to Avoid
❌Mistake: Confusing which side is opposite and which is adjacent to your angle
✅Fix: Always label your triangle clearly, marking the angle you're using
❌Mistake: Using degrees when your calculator is in radian mode
✅Fix: Make sure your calculator is set to degree mode for ACT problems
❌Mistake: Forgetting to use inverse trig functions when finding angles
✅Fix: Use sin⁻¹, cos⁻¹, or tan⁻¹ when the angle is unknown
❌Mistake: Mixing up the hypotenuse with other sides
✅Fix: Remember the hypotenuse is always opposite the right angle and is the longest side
Practice Question — Try It Yourself
A ramp makes a 12° angle with the horizontal ground. If the ramp is 20 feet long, what is the vertical height the ramp rises?
A) 4.2 feet
B) 19.5 feet
C) 96.4 feet
D) 1.7 feet
E) 6.8 feet
Show Answer
Answer: A — Using SOH: sin 12° = height/20, so height = 20 × sin 12° = 20 × 0.208 = 4.2 feet
Key Takeaways for the ACT
SOHCAHTOA helps you choose the right trig ratio — memorize it completely
Always draw and label your right triangle to visualize the problem
Use your calculator confidently since it's allowed throughout the ACT math section
Inverse trig functions (sin⁻¹, cos⁻¹, tan⁻¹) find angles when you know the sides
ACT math trigonometry problems often involve real-world applications like ladders, buildings, and ramps
Related ACT Math Topics
Strengthen your ACT math prep with these related topics:
Unit circle and trigonometric functions →
Pythagorean theorem →