🎯 Learning Objectives
- The three trig ratios: sine, cosine, tangent: and how to remember them.
- How to set up SOH-CAH-TOA for any right triangle.
- How to find a side or an angle when you know two pieces of info.
- The reciprocal trig functions and when they appear.
- 10 SAT-style practice questions.
1. Labeling the Sides
Pick one acute angle (call it θ) in your right triangle. The three sides take their names FROM that angle's perspective:
- Hypotenuse: the side opposite the right angle (always the longest).
- Opposite: the side directly across from θ.
- Adjacent: the side next to θ that is NOT the hypotenuse.
💡 Pro Tip: Always Re-label
If the question switches focus to the other acute angle, the 'opposite' and 'adjacent' sides swap! The hypotenuse never changes.
2. SOH CAH TOA
sin θ = Opposite / Hypotenuse
cos θ = Adjacent / Hypotenuse
tan θ = Opposite / Adjacent
⚡ Memory Aid
SOH = Sine: Opposite over Hypotenuse
CAH = Cosine: Adjacent over Hypotenuse
TOA = Tangent: Opposite over Adjacent
SOH-CAH-TOA is a memory aid. First label the sides relative to the chosen acute angle, then select the ratio that connects the known and unknown quantities.
3. When to Use Which Ratio
❓ Which two sides do you have / want? | ||
Opposite + Hypotenuse ▼ | Adjacent + Hypotenuse ▼ | |
Use sine | Use cosine | |
If you have OPPOSITE and ADJACENT (no hypotenuse involved), use TANGENT.
4. Solving for a Missing Side
📝 Worked Example
In a right triangle, θ = 30°, and the hypotenuse is 10. Find the side opposite to θ.
Setup: sin 30° = opposite / 10.
Since sin 30° = 1/2, we get opposite = 10 × 1/2 = 5.
5. Solving for a Missing Angle
If you know two sides but not the angle, use the inverse trig functions: sin⁻¹, cos⁻¹, tan⁻¹. (On a calculator, these are usually labeled arcsin, arccos, arctan, or marked with a ⁻¹ exponent.)
If sin θ = x, then θ = sin⁻¹(x)
6. The Reciprocal Trig Functions
Function | Definition | Reciprocal Of |
Cosecant (csc) | Hypotenuse / Opposite | Sine |
Secant (sec) | Hypotenuse / Adjacent | Cosine |
Cotangent (cot) | Adjacent / Opposite | Tangent |
💡 Pro Tip: Reciprocals Rarely Appear
On the SAT, you'll mostly see sin, cos, and tan. The reciprocals come up rarely. If you see csc, sec, or cot, just flip the corresponding main function.
7. Trig Values You Should Know Cold
Angle | sin | cos | tan |
0° | 0 | 1 | 0 |
30° | 1/2 | √3/2 | 1/√3 |
45° | √2/2 | √2/2 | 1 |
60° | √3/2 | 1/2 | √3 |
90° | 1 | 0 | undefined |
8. Strategies
⚡ Always identify hypotenuse first
Then label opposite and adjacent based on the angle.
⚡ Memorize sin/cos/tan of 0°, 30°, 45°, 60°, 90°
These exact values are useful when a question requests an exact trigonometric result.
⚡ For inverse trig, use the calculator
Set the calculator to the right mode (degrees or radians) before using it.
⚡ Use the special right triangles
30-60-90 and 45-45-90 give every standard trig value without a calculator.
⚡ For word problems, draw the triangle
Label every side and angle before writing the trig equation.
9. Summary
📌 Key Takeaways
- SOH: sin = opp / hyp.
- CAH: cos = adj / hyp.
- TOA: tan = opp / adj.
- For missing sides: set up trig equation, solve.
- For missing angles: use sin⁻¹, cos⁻¹, tan⁻¹.
- Reciprocals: csc, sec, cot: flip the main ratios.
10. Practice: 10 Questions
Q1. In a right triangle with hypotenuse 10 and opposite side 6, what is sin θ?
A) 3/5 B) 4/5 C) 5/3 D) 6/10
Hint: Opposite / hypotenuse = 6/10 = 3/5.
Q2. In the same triangle (legs 6 and 8, hyp 10), what is cos θ?
A) 3/5 B) 4/5 C) 5/3 D) 6/10
Hint: Adjacent (8) / hypotenuse (10) = 4/5.
Q3. In a right triangle with legs 3 and 4 (hypotenuse 5), tan of the angle opposite 3?
A) 3/4 B) 4/3 C) 3/5 D) 4/5
Hint: Opposite (3) / adjacent (4) = 3/4.
Q4. sin 30° = ?
A) 1/2 B) √2/2 C) √3/2 D) 1
Hint: Memorized: 30° = 1/2.
Q5. cos 60° = ?
A) 1/2 B) √2/2 C) √3/2 D) 1
Hint: Memorized.
Q6. A right triangle has hypotenuse 8 and one angle 30°. Length of the side opposite the 30°?
A) 4 B) 4√3 C) 8 D) 8√3
Hint: sin 30° × 8 = 1/2 × 8 = 4.
Q7. A ladder leans against a wall, hypotenuse 13, base 5. Height it reaches?
A) 8 B) 12 C) 13 D) 18
Hint: 5-12-13 triple.
Q8. sin θ = 7/25. cos θ?
A) 7/24 B) 24/25 C) 25/7 D) 7/25
Hint: 7-24-25 triple → adj = 24.
Q9. tan 45° = ?
A) 0 B) 1/2 C) 1 D) √2
Hint: Memorized: tan 45° = 1.
Q10. A 12-foot ramp makes a 30° angle with the ground. How high?
A) 4 B) 6 C) 6√3 D) 12
Hint: sin 30° × 12 = 6.
Answer Key & Worked Solutions
# | Answer | Reasoning |
Q1. | A) 3/5 | 6/10 = 3/5. |
Q2. | B) 4/5 | 8/10 = 4/5. |
Q3. | A) 3/4 | Opp/adj. |
Q4. | A) 1/2 | Standard value. |
Q5. | A) 1/2 | Standard value. |
Q6. | A) 4 | (1/2) × 8. |
Q7. | B) 12 | 5-12-13 triple. |
Q8. | B) 24/25 | Pythagorean triple. |
Q9. | C) 1 | Standard value. |
Q10. | B) 6 | sin 30° × 12 = 6. |