Find $\sin \theta$.
$\tan \theta=-\frac{\sqrt{5}}{4}, \cos \theta>0$
$\sin \theta=$
$\square$
(Simplify your answer, including any radicals.)
Answer (1)
We are given tan θ = − 4 5 and 0"> cos θ > 0 , which means θ is in the fourth quadrant.
Use the identity tan 2 θ + 1 = sec 2 θ to find sec θ = 4 21 .
Find cos θ = s e c θ 1 = 21 4 = 21 4 21 .
Use the identity sin 2 θ + cos 2 θ = 1 to find sin θ = − 1 − cos 2 θ = − 21 105 .
− 21 105
Explanation
Analyze the problem We are given that tan θ = − 4 5 and 0"> cos θ > 0 . We want to find sin θ . Since tan θ is negative and cos θ is positive, θ must be in the fourth quadrant, where sin θ is negative.
Find sec squared theta We know that tan 2 θ + 1 = sec 2 θ . Substituting the given value of tan θ , we have sec 2 θ = ( − 4 5 ) 2 + 1 = 16 5 + 1 = 16 5 + 16 16 = 16 21 .
Find sec theta Since 0"> cos θ > 0 , 0"> sec θ > 0 . Therefore, we take the positive square root: sec θ = 16 21 = 4 21 .
Find cos theta Now we find cos θ using the identity cos θ = s e c θ 1 :
cos θ = 4 21 1 = 21 4 = 21 4 21 .
Find sin squared theta We use the identity sin 2 θ + cos 2 θ = 1 to find sin θ :
sin 2 θ = 1 − cos 2 θ = 1 − ( 21 4 ) 2 = 1 − 21 16 = 21 21 − 21 16 = 21 5 .
Find sin theta Since sin θ < 0 in the fourth quadrant, we take the negative square root: sin θ = − 21 5 = − 21 5 = − 21 5 21 = − 21 105 .
Examples
Understanding trigonometric functions like sine, cosine, and tangent is crucial in various real-world applications. For instance, in navigation, these functions help determine the direction and distance of travel. Imagine a ship sailing at sea; by knowing the angle of its course relative to a fixed point and the distance traveled, one can calculate the ship's exact position using trigonometric relationships. Similarly, in engineering, these functions are used to analyze forces and stresses in structures, ensuring stability and safety. For example, when designing a bridge, engineers use trigonometric functions to calculate the angles and tensions in cables and supports, ensuring the bridge can withstand the intended loads. Trigonometry, therefore, provides essential tools for solving problems involving angles and distances in numerous practical scenarios.
