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Popular >

12=3sec(θ)+5csc(θ)

Solution

12 = 3 sec (θ) + 5 csc (θ)

Solution

θ = π - 0.33570 \dots + 2 πn, θ = 1.07609 \dots + 2 πn, θ = 0.65383 \dots + 2 πn, θ = - 1.39422 \dots + 2 πn

+1

Solution steps

12 = 3 sec (θ) + 5 csc (θ)

Subtract

5 csc (θ)

from both sides

3 sec (θ) = 12 - 5 csc (θ)

Square both sides

(3 sec (θ))^{2} = (12 - 5 csc (θ))^{2}

Subtract

(12 - 5 csc (θ))^{2}

from both sides

9 sec^{2} (θ) - 144 + 120 csc (θ) - 25 csc^{2} (θ) = 0

Express with sin, cos

\frac{- 25 cos ^{2} ( θ ) + 9 sin ^{2} ( θ ) + 120 cos ^{2} ( θ ) sin ( θ ) - 144 cos ^{2} ( θ ) sin ^{2} ( θ )}{cos ^{2} ( θ ) sin ^{2} ( θ )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

- 25 cos^{2} (θ) + 9 sin^{2} (θ) + 120 cos^{2} (θ) sin (θ) - 144 cos^{2} (θ) sin^{2} (θ) = 0

Rewrite using trig identities

- 25 - 110 sin^{2} (θ) + 120 sin (θ) - 120 sin^{3} (θ) + 144 sin^{4} (θ) = 0

Solve by substitution

sin (θ) \approx 0.32943 \dots, sin (θ) \approx 0.88011 \dots, sin (θ) \approx 0.60823 \dots, sin (θ) \approx - 0.98445 \dots

sin (θ) = 0.32943 \dots : θ = arcsin (0.32943 \dots) + 2 πn, θ = π - arcsin (0.32943 \dots) + 2 πn

sin (θ) = 0.88011 \dots : θ = arcsin (0.88011 \dots) + 2 πn, θ = π - arcsin (0.88011 \dots) + 2 πn

sin (θ) = 0.60823 \dots : θ = arcsin (0.60823 \dots) + 2 πn, θ = π - arcsin (0.60823 \dots) + 2 πn

sin (θ) = - 0.98445 \dots : θ = arcsin (- 0.98445 \dots) + 2 πn, θ = π + arcsin (0.98445 \dots) + 2 πn

Combine all the solutions

θ = arcsin (0.32943 \dots) + 2 πn, θ = π - arcsin (0.32943 \dots) + 2 πn, θ = arcsin (0.88011 \dots) + 2 πn, θ = π - arcsin (0.88011 \dots) + 2 πn, θ = arcsin (0.60823 \dots) + 2 πn, θ = π - arcsin (0.60823 \dots) + 2 πn, θ = arcsin (- 0.98445 \dots) + 2 πn, θ = π + arcsin (0.98445 \dots) + 2 πn

Verify solutions by plugging them into the original equation

θ = π - arcsin (0.32943 \dots) + 2 πn, θ = arcsin (0.88011 \dots) + 2 πn, θ = arcsin (0.60823 \dots) + 2 πn, θ = arcsin (- 0.98445 \dots) + 2 πn

Show solutions in decimal form

θ = π - 0.33570 \dots + 2 πn, θ = 1.07609 \dots + 2 πn, θ = 0.65383 \dots + 2 πn, θ = - 1.39422 \dots + 2 πn

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