Find the exact value of the trigonometric function at the given real number.

Question

Find the exact value of the trigonometric function at the given real number.

cos (pi/3)

sec(-pi/3)

cos (31pi/6)

sin(5pi/3)

tan(5pi/3)

sin(11pi/6)

cos(29pi/6)

Answer 1

1. cos(π/3): The cosine of π/3 (or 60 degrees) is 1/2.

2. sec(-π/3): The secant is the reciprocal of the cosine. cos(-π/3) is the same as cos(π/3), so sec(-π/3) = 1/cos(-π/3) = 2.

3. cos(31π/6): Since the period of the cosine function is 2π, cos(31π/6) = cos(31π/6 - 5π) = cos(π/6), which is √3/2.

4. sin(5π/3): Using the periodicity of the sine function, sin(5π/3) = sin(5π/3 - 2π) = sin(-π/3), which equals -√3/2.

5. tan(5π/3): The tangent function is periodic with a period of π, so tan(5π/3) = tan(5π/3 - 2π) = tan(-π/3), which equals -√3.

6. sin(11π/6): Using periodicity, sin(11π/6) = sin(11π/6 - 2π) = sin(-π/6), which is -1/2.

7. cos(29π/6): Similar to the other calculations, cos(29π/6) = cos(29π/6 - 4π) = cos(π/6), which is √3/2.