General solution for sine
Solve sin(θ)=k where -1≦k≦1
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
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
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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