What is the exact value of $\cos \left(195^{\circ}\right) ?$
A. $\frac{\sqrt{6}-\sqrt{2}}{4}$
B. $\frac{\sqrt{6}+\sqrt{2}}{4}$
C. $\frac{-\sqrt{6}-\sqrt{2}}{4}$
D. $\frac{-\sqrt{6}+\sqrt{2}}{4}$
Answer (1)
Express 19 5 ∘ as a sum of two angles: 19 5 ∘ = 15 0 ∘ + 4 5 ∘ .
Apply the cosine sum formula: cos ( A + B ) = cos ( A ) cos ( B ) − sin ( A ) sin ( B ) .
Substitute the known values: cos ( 19 5 ∘ ) = ( − 2 3 ) ( 2 2 ) − ( 2 1 ) ( 2 2 ) .
Simplify to find the exact value: 4 − 6 − 2 .
Explanation
Express 195 degrees as a sum of known angles We are asked to find the exact value of cos ( 19 5 ∘ ) . We can express 19 5 ∘ as the sum of two angles whose cosine and sine values are known. A good choice is 19 5 ∘ = 15 0 ∘ + 4 5 ∘ .
Apply the cosine sum formula Apply the cosine sum formula: cos ( A + B ) = cos ( A ) cos ( B ) − sin ( A ) sin ( B ) . In our case, A = 15 0 ∘ and B = 4 5 ∘ .
State known cosine and sine values We know that cos ( 15 0 ∘ ) = − 2 3 , cos ( 4 5 ∘ ) = 2 2 , sin ( 15 0 ∘ ) = 2 1 , and sin ( 4 5 ∘ ) = 2 2 .
Substitute values into the formula Substitute these values into the cosine sum formula: cos ( 19 5 ∘ ) = cos ( 15 0 ∘ ) cos ( 4 5 ∘ ) − sin ( 15 0 ∘ ) sin ( 4 5 ∘ ) cos ( 19 5 ∘ ) = ( − 2 3 ) ( 2 2 ) − ( 2 1 ) ( 2 2 )
Simplify the expression Simplify the expression: cos ( 19 5 ∘ ) = − 4 6 − 4 2 = 4 − 6 − 2
State the final answer The exact value of cos ( 19 5 ∘ ) is 4 − 6 − 2 .
Examples
Understanding trigonometric functions like cosine is crucial in fields like physics and engineering. For instance, when analyzing the motion of a pendulum, the cosine function helps describe the horizontal displacement of the pendulum bob over time. Similarly, in electrical engineering, cosine functions are used to model alternating current (AC) waveforms, allowing engineers to predict and control the behavior of electrical circuits.
