### Inverse Trigonometric Functions Exercise 01

## Question:

Find the principal value of cot^{−1}(√3).

## Answer:

Answer:
Step 1: Recall that cot^{−1}(x) is the inverse of cot(x).

Step 2: Recall that cot(x) = cos(x)/sin(x).

Step 3: Substitute √3 for x in the equation cot(x) = cos(x)/sin(x).

Step 4: Solve for cos(x)/sin(x) = √3.

Step 5: Recall that cos(x) = sin(90° - x).

Step 6: Substitute 90° - x for cos(x) in the equation sin(90° - x)/sin(x) = √3.

Step 7: Solve for sin(90° - x)/sin(x) = √3.

Step 8: Recall that sin(90° - x) = cos(x).

Step 9: Substitute cos(x) for sin(90° - x) in the equation cos(x)/sin(x) = √3.

Step 10: Solve for cos(x)/sin(x) = √3.

Step 11: Recall that cot-1(x) is the inverse of cot(x).

Step 12: Substitute the value obtained in Step 10 for x in the equation cot-1(x).

Step 13: The principal value of cot^{−1}(√3) is 60°.

## Question:

Find the principal value of sec^{−1}(2/√3).

## Answer:

Step 1: Find the inverse of secant.

The inverse of secant is cosecant, which is written as csc or cosec.

Step 2: Substitute the given value in the cosecant equation.

csc^−1(2/√3)

Step 3: Solve for the principal value.

The principal value of csc^{−1}(2/√3) is 60°.

## Question:

Find the principal value of cos^{−1}(√3/2)

## Answer:

Answer: Step 1: Convert √3/2 into a decimal by dividing 3 by 2.

Answer: 1.5

Step 2: Calculate the inverse cosine of 1.5.

Answer: The principal value of cos^{−1}(√3/2) is 60°.

## Question:

Find the principal value of tan^{−1}(−√3).

## Answer:

Step 1: Recall that tan^{−1}(x) is the inverse of the tangent function, which is equal to the angle whose tangent is equal to x.

Step 2: Since tan^{−1}(−√3) is the angle whose tangent is equal to -√3, we can use a trigonometric identity to solve for the angle.

Step 3: The identity we will use is tan(α) = -√3, where α is the angle we are trying to find.

Step 4: Solving for α, we get α = tan^{−1}(-√3).

Step 5: The principal value of tan^{−1}(−√3) is -60°.

## Question:

Find the principal value of cos^{−1}(−1/√2)

## Answer:

Answer:

- cos^-1(x) is the inverse cosine function, which is also known as the arccosine function.
- To find the principal value of cos
^{−1}(-1/√2), we need to calculate the arccosine of -1/√2. - Using a calculator, we can find that the arccosine of -1/√2 is equal to 3π/4.
- Therefore, the principal value of cos
^{−1}(-1/√2) is 3π/4.

## Question:

Find the principal value of cosec^{−1}(2).

## Answer:

Step 1: Recall that cosec^{−1}(x) is the inverse of cosec(x).

Step 2: Recall that cosec(x) = 1/sin(x).

Step 3: Set 1/sin(x) = 2.

Step 4: Solve for x by taking the inverse sine of both sides.

Step 5: The principal value of cosec^{−1}(2) is π/3 radians.

## Question:

Find the value of tan^{−1}(1)+cos^{−1}(−1/2)+sin^{−1}(−1/2).

## Answer:

Step 1: tan^{−1}(1) = π/4

Step 2: cos^{−1}(−1/2) = 3π/4

Step 3: sin^{−1}(−1/2) = -π/2

Step 4: Add the three values together: π/4 + 3π/4 + (-π/2) = π/2

Therefore, the value of tan^{−1}(1)+cos^{−1}(−1/2)+sin^{−1}(−1/2) is π/2.

## Question:

Find the principal value of cos^{−1}(−1/2).

## Answer:

Step 1: Recall that the principal value of cos^{−1}(x) is the angle θ in the interval [0, 2π] such that cos(θ) = x.

Step 2: We are given that x = −1/2.

Step 3: Solve the equation cos(θ) = −1/2 to find θ.

Step 4: Using a calculator or a table of values, we find that θ = 2.094395102393195 radians.

Step 5: Therefore, the principal value of cos^{−1}(−1/2) is 2.094395102393195 radians.

## Question:

Find the principal value of sin^{−1}(−1/2)

## Answer:

Step 1: Recall that sin^{−1}(x) is the inverse of the sine function, which is also known as the arcsin function.

Step 2: Use the arcsin function to find the principal value of -1/2.

Step 3: The principal value of sin^{−1}(-1/2) is -π/6.

## Question:

Find the principal value of cosec^{−1}(−√2)

## Answer:

Answer:
Step 1: Recall that cosec^{−1}(x) = arcsin(1/x)

Step 2: Substitute -√2 for x in the equation:
cosec^{−1}(-√2) = arcsin(1/(-√2))

Step 3: Simplify the expression:
cosec^{−1}(-√2) = arcsin(-1/√2)

Step 4: Use a calculator to find the principal value of arcsin(-1/√2):
cosec^{−1}(-√2) = -π/4

## Question:

Find the value of cos^{−1}(1/2)+2sin^{−1}(1/2)

## Answer:

Given,
cos^{−1}(1/2) + 2sin^{−1}(1/2)

Step 1: Calculate the value of cos^−1(1/2)

cos^{−1}(1/2) = 60°

Step 2: Calculate the value of sin^{−1}(1/2)

sin^{−1}(1/2) = 30°

Step 3: Substitute the values in the given equation.

60° + 2(30°) = 120°

Hence, the value of cos^{−1}(1/2) + 2sin^{−1}(1/2) = 120°.

## Question:

Find the value of tan^{−1} √3 −sec^{−1}(−2) is equal to
A π
B −π/3
C 3π
D 2π/3

## Answer:

Answer: B -π/3

## Question:

Find the principal value of tan^{−1}(−1).

## Answer:

Step 1: Recall that the principal value of tan^{−1}(x) is the angle measured in the counterclockwise direction from the positive x-axis to the ray from the origin that intersects the point (x,y).

Step 2: In this case, x = -1, so the principal value of tan^{−1}(-1) is the angle measured in the counterclockwise direction from the positive x-axis to the ray from the origin that intersects the point (-1,y).

Step 3: Since y = -1, the principal value of tan^{−1}(-1) is the angle measured in the counterclockwise direction from the positive x-axis to the ray from the origin that intersects the point (-1,-1).

Step 4: Therefore, the principal value of tan^{−1}(-1) is π radians or 180°.