Q. The angle θ1 is located in Quadrant IV, and cos(θ1)=53.What is the value of sin(θ1) ? Express your answer exactly.sin(θ1)=
Given Information: We are given that cos(θ1)=53 and that θ1 is in Quadrant IV. In Quadrant IV, cosine is positive and sine is negative. We can use the Pythagorean identity sin2(θ)+cos2(θ)=1 to find the value of sin(θ1).
Solving for sin2(θ): First, we will solve for sin2(θ1). We know that cos2(θ1)=(53)2. cos2(θ1)=259
Using Pythagorean Identity: Now, we use the Pythagorean identity to find sin2(θ1). sin2(θ1)=1−cos2(θ1) sin2(θ1)=1−259 sin2(θ1)=2525−259 sin2(θ1)=2516
Finding sin(θ): Next, we take the square root of both sides to find sin(θ1). Since θ1 is in Quadrant IV, we must take the negative square root because sine is negative in this quadrant.sin(θ1)=−2516sin(θ1)=−54
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