Question

Write the following expression as the sine, cosine, or tangent of a double angle. Then find the exact value of the expression.

2 sin 105° cos 105°
Write the following expression as the sine, cosine, or tangent of a double angle. Select the correct choice below and fill in the answer box to complete your choice. O (Simplify your answer. Type your answer in degrees. Use integers or decimals for any numbers in the expression.)
A. 2 sin 105° cos 105º = cos ___º
B. 2 sin 105° cos 105º = sin ___º
C. 2 sin 105° cos 105º = tan___º
The exact value of the given expression is ___º (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression. Rationalize the denominator.)

Answer 1

The given expression 2 sin 105° cos 105° can be written as sin(210°) using the double angle identity for sine, and the exact value of sin(210°) is -1/2.

Step-by-step explanation:

To express the expression 2 sin 105° cos 105° as the sine, cosine, or tangent of a double angle, we can use the identity for sin(2θ) = 2 sin(θ) cos(θ). Therefore, applying this identity to our problem:

2 sin 105° cos 105° = sin(2 × 105°) = sin(210°)

The correct choice is therefore option B. 2 sin 105° cos 105° = sin 210°.

To find the exact value of sin(210°), we use the fact that 210° is in the third quadrant, where sine is negative. Since 210° is 30° more than 180°, we can use the reference angle of 30° within the unit circle concept:

sin(210°) = -sin(30°)

The exact value of sin(30°) is 1/2, thus the expression simplifies to:

sin(210°) = -1/2