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:

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