What is the value of [tex]$\cos \left(150^{\circ}\right) ?[/tex]
A. [tex]$\frac{\sqrt{3}}{2}[/tex]
B. [tex]$-\frac{\sqrt{3}}{2}[/tex]
C. [tex]$\frac{1}{2}[/tex]
D. [tex]$-\frac{1}{2}[/tex]
Answer (2)
Recognize that 15 0 ∘ is in the second quadrant, where cosine is negative.
Express 15 0 ∘ as 18 0 ∘ − 3 0 ∘ .
Use the identity cos ( 18 0 ∘ − x ) = − cos ( x ) , so cos ( 15 0 ∘ ) = − cos ( 3 0 ∘ ) .
Since cos ( 3 0 ∘ ) = 2 3 , then cos ( 15 0 ∘ ) = − 2 3 .
Explanation
Problem Analysis The problem asks us to find the value of cos ( 15 0 ∘ ) . We need to determine which of the given options is correct.
Quadrant Analysis We know that 15 0 ∘ lies in the second quadrant. In the second quadrant, the cosine function is negative. We can express 15 0 ∘ as 18 0 ∘ − 3 0 ∘ .
Applying Trigonometric Identity Using the identity cos ( 18 0 ∘ − x ) = − cos ( x ) , we have: cos ( 15 0 ∘ ) = cos ( 18 0 ∘ − 3 0 ∘ ) = − cos ( 3 0 ∘ )
Evaluating Cosine We know that cos ( 3 0 ∘ ) = 2 3 . Therefore, cos ( 15 0 ∘ ) = − 2 3 .
Final Answer The value of cos ( 15 0 ∘ ) is − 2 3 .
Examples
Understanding trigonometric functions like cosine is crucial in many real-world applications. For instance, in physics, when analyzing projectile motion, the cosine function helps determine the horizontal component of the initial velocity. Similarly, in engineering, calculating the forces acting on structures often involves using cosine to resolve vectors into their components. Even in music, the cosine function is used to model sound waves, helping us understand the properties of different tones and harmonies.
The cosine of 15 0 ∘ is − 2 3 , which we find by analyzing that it is in the second quadrant where cosine is negative and using the identity for cosine with an angle expressed as 18 0 ∘ − x . Therefore, the correct answer is option B: − 2 3 .
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