Inverse Trig Worksheet Algebra 2 Without Calculator

Introduction

Inverse trigonometric functions are essential tools in Algebra 2 that allow us to find angles when we know the ratio of sides in a right triangle. Unlike the basic trigonometric functions (sine, cosine, tangent), which take an angle and return a ratio, inverse trigonometric functions take a ratio and return an angle.

This worksheet focuses on three main inverse trigonometric functions:

• arcsin (or sin⁻¹) - Finds the angle whose sine is the given value
• arccos (or cos⁻¹) - Finds the angle whose cosine is the given value
• arctan (or tan⁻¹) - Finds the angle whose tangent is the given value

All problems in this worksheet can be solved using the unit circle and reference angles without a calculator.

Inverse Sine (arcsin)

The arcsin function, written as sin⁻¹(x), returns the angle θ in radians where -π/2 ≤ θ ≤ π/2 and sin(θ) = x.

Key Points

• The output of arcsin is always in the range [-π/2, π/2]
• For any angle θ in this range, sin(θ) is one-to-one (injective)
• The arcsin function is undefined for values outside [-1, 1]

Example Problem

Find θ = sin⁻¹(0.5)

Solution:

1. Recall that sin(π/6) = 0.5
2. Since π/6 is within the range of arcsin, θ = π/6 radians (30°)

Inverse Cosine (arccos)

The arccos function, written as cos⁻¹(x), returns the angle θ in radians where 0 ≤ θ ≤ π and cos(θ) = x.

Key Points

• The output of arccos is always in the range [0, π]
• For any angle θ in this range, cos(θ) is one-to-one
• The arccos function is undefined for values outside [-1, 1]

Example Problem

Find θ = cos⁻¹(-0.5)

Solution:

1. Recall that cos(2π/3) = -0.5
2. Since 2π/3 is within the range of arccos, θ = 2π/3 radians (120°)

Inverse Tangent (arctan)

The arctan function, written as tan⁻¹(x), returns the angle θ in radians where -π/2 < θ < π/2 and tan(θ) = x.

Key Points

• The output of arctan is always in the range (-π/2, π/2)
• For any angle θ in this range, tan(θ) is one-to-one
• The arctan function is defined for all real numbers

Example Problem

Find θ = tan⁻¹(1)

Solution:

1. Recall that tan(π/4) = 1
2. Since π/4 is within the range of arctan, θ = π/4 radians (45°)

Practice Problems

Solve the following inverse trigonometric problems without using a calculator. Show your work for each problem.

Tips for Success

1. Understand the range of each function:
   • arcsin: -π/2 to π/2
   • arccos: 0 to π
   • arctan: -π/2 to π/2
2. Use reference angles: Remember the common angles and their sine, cosine, and tangent values.
3. Check your work: Verify that the sine, cosine, or tangent of your answer equals the given value.
4. Practice regularly: The more you work with inverse trigonometric functions, the more comfortable you'll become with them.