Question

The terminal side of angle θ intersects the unit circle in the first quadrant at (16/19,y). What are the exact values of sinθ and cosθ?

Answer 1

The exact value of sin θ  and cos θ  is: sinθ =(√105)/19 and cosθ =16/19

The terminal side of an angle is the angle at the standard position. The terminal side of an angle θ drawn in angle standard position is the side which isn't the initial side.

Given that the point of intersection of the terminal side of θ an the unit circle is:

(16/19,y)

The exact value of sin θ  and cos θ  is: sinθ =(√105)/19 and cosθ =16/19

The given parameters is represented by:

(16/19,y)

This means that :

(cosθ ,sinθ )=(16/19,y)

Using the following trigonometric identity:

cos²θ +sinθ =1

We have:

(16/19)²+y²=1

Expand fraction:

(256/361)+y²=1

Collect like terms:

y²=1-(256/361)

Take LCM:

y²=(361-256)/361

y²=105/361

Take square roots

y=(√105)/19

Substitute value for y in

(cosθ ,sinθ )=(16/19,y)

By comparison:

cosθ =16/19

sinθ =(√105)/19

So the exact value of sin θ  and cos θ  is: sinθ =(√105)/19 and cosθ =16/19

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