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Question

Consider <A such that cos A= 12/13

a) in which quadrant(s) could this angle exist?

b) if the sin A is negative in which quadrant is the angle?

c) write exact expression for the other two primary trigonometric ratios for the angle.

Answer 1

The trigonometric function value is cos(A) = 12/13.

(a).

The value of cosine function is positive in first and fourth quadrant, so the angle A lies in either first quadrant or fourth quadrant.

(b).

The value of sine function is negative in third and fourth quadrant, so the angle A lies in either third quadrant or fourth quadrant.

The value of cosine function is positive in first and fourth quadrant, so the angle A lies in either first quadrant or fourth quadrant.

Therefore, sin(A) is negative and cos(A) is positive in fourth quadrant. So the angle A lies in fourth quadrant.

(c).

Using the Pythagorean identity : sin²(θ) + cos²(θ) = 1, we obtain

sin²(A) + (12/13)² = 1

sin²(A) = 1 - 144/169 = 25/169.

Because sin(θ) < 0 in quadrant IV, we can use the negative root to obtain

sin(A) = - √25/√169 = - 5/13.

Using the trigonometric identity : tan(A) = sin(A)/cos(A), we obtain

tan(A) = (- 5/13) / (12/13) = - 5/12.

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