General Solutions of Trigonometric Equations in Degrees – Sine, Cosine, Tangent (A-level / Further Maths GCSE)

This guide explains how to solve general solutions of trigonometric equations in degrees for sine, cosine, and tangent, using the PV method, with step-by-step examples suitable for A-level and Further Maths GCSE students.

How to Solve General Trigonometric Equations in Degrees

Trigonometric functions repeat at regular intervals, so there’s usually more than one solution. The Principal Value (PV) method is a simple and visual way to find all solutions.

Steps:

1. Find the Principal Value (PV) using the inverse function (sin⁻¹, cos⁻¹, tan⁻¹).

2. Use the general solution formula for that function.

3. Substitute integer values of n to find all solutions within the required interval.

Tip: The PV is the first angle you get from the inverse function. The second solution comes from the “other” quadrant formula for sine or cosine.

General Solutions for Sine Equations (Degrees)

Solve sin(θ)=k where -1≦k≦1

Step 1: Find Principal Value (PV)

PV = sin⁻¹(k)

Graph of y=sin⁡x showing intersections with y=k to identify the principal value (PV) and the second solution.

Step 2: General Solution

---

Example 1

Solve for 0≦θ≦360° the equation

Taking inverse sine on both sides we get,

Now write down the general solution before trying to solve for θ.

n=0 ⇒ θ=85°

n=1 ⇒ θ=145°, 265°

n=2 ⇒ θ=325°

---

Example 2

Solve for 0≦x≦180° the equation

Rearranging and taking inverse sine we get,

Using the sine formula,

n=0 ⇒ θ=13.9°, 46.1°

n=1 ⇒ θ=133.9°, 166.1°

N.B. We reject n=2 since this would give solutions greater than 180°.

---

Example 3

Solve the equation

Using the sine formula,

Now substitute n with any integer to find specific solutions.

---

Example 4

Solve for 0≦θ≦360° the equation

Using the sine formula,

n=0 ⇒ x=22.5°, 67.5°, 112.5°

n=1 ⇒ x=202.5°, 247.5°, 157.5°, 292.5°

n=2 ⇒ x=337.5°

---

General Solutions for Cosine Equations (Degrees)

Solve cos(θ)=k where -1≦k≦1

Step 1: Find Principal Value (PV)

PV = cos⁻¹(k)

Graph of y=cos⁡x showing intersections with y=k to identify PV and the second solution.

Step 2: General Solution

---

Example 1

Solve for 0≦θ≦180° the equation

Using the cosine formula,

n=0 ⇒ x=10°, 50°

n=1 ⇒ x=130°, 110°, 170°, 70°

---

Example 2

Solve for -180°≦x≦180° the equation

Using the cosine formula,

n=0 ⇒ x=47.5°, -87.5°

n=1 ⇒ x=92.5°

n=-1 ⇒ x=-132.5°

---

General Solutions for Tangent Equations (Degrees)

Solve tan(θ)=k where k∈ℝ

Step 1: Find Principal Value (PV)

PV = tan⁻¹(k)

Graph of y=tan⁡x showing intersections with y=k to identify PV and repeated solutions.

Step 2: General Solution

Tip: Tangent repeats every 180°, so only one formula is needed.

---

Example

Solve for 0≦x≦360° the equation

Using the tan formula,

n=0 ⇒ x=63.4°

n=1 ⇒ x=243.4°, 108.4°

n=2 ⇒ x=288.4°

Conclusion

Understanding general solutions of trigonometric equations in degrees for sine, cosine, and tangent is essential for A-level and Further Maths GCSE students. Using the PV method and following the step-by-step approach makes it easier to find all solutions accurately.

If you want to see different examples worked out in radians, check out my related post: General Solutions of Trigonometric Equations (Radians). Both posts complement each other and provide a full understanding of trigonometric equations in both degrees and radians.