Question

Fill in the blanks to make each statement true. arcsin(_______) = arccos( _______ ) = arctan( _______) = 60°

Answer 1

To make each statement true, the values to fill in the blanks are √3/2 for arcsin, 1/2 for arccos, and √3 for arctan, as these values would result in a 60° angle for each respective function.

Step-by-step explanation:

The question asks to fill in the blanks to make each statement true: arcsin(_______) = arccos(_______) = arctan(_______) = 60°. Let's find the values that satisfy the trigonometric equations.

To solve this, recall that sine, cosine, and tangent are functions of angles, associated with the ratios of sides in a right triangle. For an angle of 60° in a standard right triangle (where the angles are 30°, 60°, and 90°), the relationships are:

sin(60°) is √3/2

cos(60°) is 1/2

tan(60°) is √3

However, to make the arcsin, arccos, and arctan functions all equal to the same angle, we'll need to find a value that produces the same angle when each of these inverse trigonometric functions is applied. Noting that the sine and cosine of an angle are the cos and sin (respectively) of its complement and that the tangent of an angle is the reciprocal of the tangent of its complementary angle:

arcsin(√3/2) = 60°

arccos(1/2) = 60°

arctan(√3) = 60

Answer 2

The blanks can be filled with (
`(√(3) )/(2)`), (
`(1)/(2)`), and (
`√(3)`).

To find the missing values that make each statement true, let's consider the relationships between trigonometric functions and their inverses :

arcsin
`(√(3) )/(2)` = arccos
`(1)/(2)` = arctan
`√(3)`= 60°

In this case :

- arcsin
`((√(3) )/(2))`is the angle whose sine is
`((√(3) )/(2))`which is (
`(\pi )/(3)`) or (60°).

- arccos (
`(1)/(2)`) is the angle whose cosine is (
`(\pi )/(3)`), which is (
`(\pi )/(3)`) or (60°).

- arctan (
`√(3)`) is the angle whose tangent is (
`√(3)`), which is (
`(\pi )/(3)`) or (60°).