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[tex]\sin \frac{2 \pi}{5} \cos \frac{\pi}{10}+\cos \frac{2 \pi}{5} \sin \frac{\pi}{10}=[/tex]
[tex]\cos 13^{\circ} \cos 47^{\circ}-\sin 13^{\circ} \sin 47^{\circ}=[/tex]
[tex]\sin \frac{2 \pi}{5} \cos \frac{8 \pi}{5}+\cos \frac{2 \pi}{5} \sin \frac{8 \pi}{5}=[/tex]
Answer (1)
The first expression uses the sine addition formula and simplifies to sin ( 2 π ) , which equals 1.
The second expression uses the cosine addition formula and simplifies to cos ( 6 0 ∘ ) , which equals 2 1 .
The third expression uses the sine addition formula and simplifies to sin ( 2 π ) , which equals 0.
The values of the three expressions are 1 , 2 1 , 0 .
Explanation
Problem Analysis We are given three trigonometric expressions to evaluate. We will use trigonometric identities to simplify each expression and then find its value.
Evaluate Expression 1 The first expression is sin 5 2 π cos 10 π + cos 5 2 π sin 10 π . This is in the form of the sine addition formula: sin ( a + b ) = sin a cos b + cos a sin b . Therefore, we can rewrite the expression as sin ( 5 2 π + 10 π ) = sin ( 10 4 π + 10 π ) = sin ( 10 5 π ) = sin ( 2 π ) . Since sin ( 2 π ) = 1 , the value of the first expression is 1.
Evaluate Expression 2 The second expression is cos 1 3 ∘ cos 4 7 ∘ − sin 1 3 ∘ sin 4 7 ∘ . This is in the form of the cosine addition formula: cos ( a + b ) = cos a cos b − sin a sin b . Therefore, we can rewrite the expression as cos ( 1 3 ∘ + 4 7 ∘ ) = cos ( 6 0 ∘ ) . Since cos ( 6 0 ∘ ) = 2 1 , the value of the second expression is 2 1 .
Evaluate Expression 3 The third expression is sin 5 2 π cos 5 8 π + cos 5 2 π sin 5 8 π . This is in the form of the sine addition formula: sin ( a + b ) = sin a cos b + cos a sin b . Therefore, we can rewrite the expression as sin ( 5 2 π + 5 8 π ) = sin ( 5 10 π ) = sin ( 2 π ) . Since sin ( 2 π ) = 0 , the value of the third expression is 0.
Final Answer Therefore, the values of the three expressions are 1, 2 1 , and 0, respectively.
Examples
Trigonometric identities are useful in many areas of physics and engineering, such as analyzing alternating current circuits or studying the motion of waves. For example, when analyzing the superposition of two waves with different phases, trigonometric identities can help simplify the resulting expression and determine the amplitude and phase of the combined wave. These calculations are crucial in designing efficient communication systems and understanding wave phenomena.
