Reference angles separate two decisions in a trigonometric equation: the size of the basic angle and the quadrants in which the required function has the correct sign. General solutions then capture the repeating pattern beyond one cycle.
1. What is a reference angle?
A reference angle is the acute angle between the terminal arm and the horizontal axis. For 150°, the reference angle is 30°. For 225°, it is 45°.
The reference angle determines the magnitude of familiar exact trig values; the quadrant determines the sign.
2. Quadrant signs
- Sine is positive in quadrants I and II.
- Cosine is positive in quadrants I and IV.
- Tangent is positive in quadrants I and III.
The reciprocal functions share the sign of the function they reciprocate where defined.
3. Worked sine example
Solve sin x=−1/2 for 0°≤x<360°. The reference angle is 30°. Sine is negative in quadrants III and IV.
Therefore x=210° or 330°.
4. Worked cosine example
Solve cos x=√2/2 for 0≤x<2π. The reference angle is π/4. Cosine is positive in quadrants I and IV.
Hence x=π/4 or 7π/4.
5. Worked tangent example
Solve tan x=−√3 for 0°≤x<360°. The reference angle is 60°. Tangent is negative in quadrants II and IV.
Thus x=120° or 300°.
6. General sine solutions
If sin x=sin α, then a useful degree form is
x=α+360°n or x=180°−α+360°n, n an integer.
In radians, replace 360° by 2π and 180° by π.
7. General cosine solutions
If cos x=cos α, then
x=2πn±α in radians, or x=360°n±α in degrees.
8. General tangent solutions
If tan x=tan α, then
x=α+nπ in radians, or x=α+180°n in degrees.
Tangent needs only one repeating family because its fundamental period is π.
9. General solutions must respect transformed inputs
For sin(2x)=sin α, write the general solutions for 2x first, then divide the complete expressions by 2. Do not divide only the reference angle and forget the periodic term.
10. Principal values are not general solutions
An inverse trig key returns one principal value. A general solution describes every angle satisfying the equation. These have different jobs and should not be confused.
11. Practice
- Find the reference angle of 140°.
- Solve sin x=−√3/2 for 0°≤x<360°.
- Solve cos x=−√2/2 for 0≤x<2π.
- Write the general solution of tan x=1 in radians.
- Write general solutions for sin x=sin(π/6).
12. Answers
- 40°.
- 240°,300°.
- 3π/4,5π/4.
- x=π/4+nπ, n∈ℤ.
- x=π/6+2πn or x=5π/6+2πn, n∈ℤ.
Continue with Solving Trigonometric Equations Over a Given Interval or return to the Additional Mathematics Hub.