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General Solutions of Trigonometric Equations (Degrees)

Updated: Apr 3, 2023

General solution for sine

Solve sin(θ)=k where -1≦k≦1

Graph of y=sin(x) intersecting the line y=k

PV = Principal Value = sin⁻¹(k)

 

Example

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

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

Solve the equation

Using the sine formula,

Now substitute n with any integer to find specific solutions.

 

Example

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 solution for cosine

Solve cos(θ)=k where -1≦k≦1

Graph of y=cos(x) intersecting the line y=k

PV = cos⁻¹(k)

 

Example

Solve for 0≦θ≦180° the equation

Using the cosine formula,

n=0 ⇒ x=10°, 50°

n=1 ⇒ x=130°, 110°, 170°, 70°

 

Example

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 solution for tan

Solve tan(θ)=k where k∈ℝ

Graph of y=tan(x) intersecting the line y=k

PV = tan⁻¹(k)

 

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°

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