Question

Solve for all values of theda to the nearest 10th of a degree in the interval zero greater than or equal to Theta greater than or equal to 360 cosecant to the second theda equal six cosecant theta +7

Answer 1

• 8.2°
• 171.8°
• 270°

Explanation:

You want the solutions to the trig equation csc(θ)² = 6csc(θ) +7 on the interval [0°, 360°].

Factored

We can write this equation in standard form, then factor it:

csc(θ)² -6csc(θ) -7 = 0

(csc(θ) -7)(csc(θ) +1) = 0

csc(θ) = 7 or -1

Trig identity

sin(θ) = 1/csc(θ) = 1/7 or -1

There are two values of θ in the range that have a sine of 1/7: 8.2° and 171.8°.

There is one value of θ such that sin(θ) = -1: θ = 270°.

The solutions are 8.2°, 171.8°, and 270°.

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Additional comment

Some calculators may be able to tell you the value of csc^-1(x) directly, without converting to sin^-1(1/x).

The second attachment shows a graphing calculator solution for csc(θ)² -6csc(θ) -7 = 0. (horizontal axis is degrees)

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