What is the exact value of [tex]$\cos \frac{7 \pi}{12}$[/tex]?
A. [tex]$\frac{\sqrt{6}-\sqrt{2}}{4}$[/tex]
B. [tex]$\frac{\sqrt{6}+\sqrt{2}}{4}$[/tex]
C. [tex]$\frac{2 \sqrt{2}-\sqrt{6}}{4}$[/tex]
D. [tex]$\frac{\sqrt{2}-\sqrt{6}}{4}$[/tex]
Answer (1)
Express 12 7 π as a sum of two known angles: 4 π + 3 π .
Apply the cosine addition formula: cos ( a + b ) = cos a cos b − sin a sin b .
Substitute the values of cos 4 π , cos 3 π , sin 4 π , and sin 3 π .
Simplify the expression to get the final answer: 4 2 − 6 .
Explanation
Problem Analysis We are asked to find the exact value of cos 12 7 π . We can express 12 7 π as a sum of two common angles for which we know the cosine and sine values.
Expressing the Angle as a Sum We can write 12 7 π = 12 3 π + 12 4 π = 4 π + 3 π .
Applying Cosine Addition Formula Now we use the cosine addition formula: cos ( a + b ) = cos a cos b − sin a sin b . Therefore, cos 12 7 π = cos ( 4 π + 3 π ) = cos 4 π cos 3 π − sin 4 π sin 3 π .
Substituting Known Values We know that cos 4 π = 2 2 , cos 3 π = 2 1 , sin 4 π = 2 2 , and sin 3 π = 2 3 . Substituting these values into the equation, we get: cos 12 7 π = 2 2 ⋅ 2 1 − 2 2 ⋅ 2 3 = 4 2 − 4 6 = 4 2 − 6 .
Final Answer Thus, the exact value of cos 12 7 π is 4 2 − 6 .
Examples
Understanding trigonometric functions like cosine is crucial in fields like physics and engineering. For instance, when analyzing alternating current (AC) circuits, the voltage and current waveforms are often modeled using sinusoidal functions. Calculating the cosine of specific angles, such as 12 7 π , helps engineers determine the instantaneous voltage or current at a particular point in time, which is essential for designing and troubleshooting electrical systems. This ensures accurate and efficient performance of electronic devices and power grids.
