$\cos (\theta)=\frac{\sqrt{2}}{2}$, and $\frac{3 \pi}{2}<\theta<2 \pi$, evaluate $\sin (\theta)$ and $\tan (\theta)$.
$\sin (\theta)=?$
$\tan (\theta)=$ $\square$
Answer (1)
Use the identity sin 2 ( θ ) + cos 2 ( θ ) = 1 to find sin ( θ ) .
Since θ is in the fourth quadrant, sin ( θ ) is negative, so sin ( θ ) = − 2 2 .
Use the identity tan ( θ ) = c o s ( θ ) s i n ( θ ) to find tan ( θ ) .
tan ( θ ) = − 1 , so the final answers are sin ( θ ) = − 2 2 and tan ( θ ) = − 1 . sin ( θ ) = − 2 2 , tan ( θ ) = − 1
Explanation
Analyze the problem and given data We are given that cos ( θ ) = 2 2 and 2 3 π < θ < 2 π . This means that θ lies in the fourth quadrant. In the fourth quadrant, cosine is positive, and sine is negative. We need to find the values of sin ( θ ) and tan ( θ ) .
Find sin ( θ ) We know the trigonometric identity sin 2 ( θ ) + cos 2 ( θ ) = 1 . We can use this to find sin ( θ ) . Substituting the given value of cos ( θ ) , we get: sin 2 ( θ ) + ( 2 2 ) 2 = 1 sin 2 ( θ ) + 4 2 = 1 sin 2 ( θ ) = 1 − 2 1 sin 2 ( θ ) = 2 1 Taking the square root of both sides, we get: sin ( θ ) = ± 2 1 = ± 2 2 Since θ is in the fourth quadrant, sin ( θ ) is negative. Therefore, sin ( θ ) = − 2 2
Find tan ( θ ) Now we can find tan ( θ ) using the identity tan ( θ ) = c o s ( θ ) s i n ( θ ) . Substituting the values of sin ( θ ) and cos ( θ ) , we get: tan ( θ ) = 2 2 − 2 2 = − 1
Final Answer Therefore, sin ( θ ) = − 2 2 and tan ( θ ) = − 1 .
Examples
Understanding trigonometric functions and their values in different quadrants is crucial in various fields like physics and engineering. For instance, when analyzing the motion of a pendulum or the trajectory of a projectile, knowing the sine, cosine, and tangent of angles helps determine the position and velocity of the object at any given time. Similarly, in electrical engineering, these functions are used to describe alternating current and voltage.
