Using SOH CAH TOA to Find Missing Sides and Angles

Finding Missing Sides

Key Steps

1. Identify the sides involved: Determine which sides (opposite, adjacent, hypotenuse) are known and unknown.
2. Select the appropriate equation: Use sine, cosine, or tangent based on the identified sides.
3. Rearrange the equation: Solve for the unknown side.

Example Problems

Problem 1

• Known: Hypotenuse (13), Angle (42 degrees)
• Unknown: Opposite (X)
• Equation: \sin(42) = \frac{X}{13}
• Solution: X = 13 \times \sin(42) ≈ 8.7

Problem 2

• Known: Adjacent (10), Angle (37 degrees)
• Unknown: Opposite (Y)
• Equation: \tan(37) = \frac{Y}{10}
• Solution: Y = 10 \times \tan(37) ≈ 7.5

Problem 3

• Known: Adjacent (13), Angle (41 degrees)
• Unknown: Hypotenuse (Q)
• Equation: \cos(41) = \frac{13}{Q}
• Solution: Q = \frac{13}{\cos(41)} ≈ 17.2

Key Concepts

• Equations: Use the correct trigonometric function based on the sides known.
• Rearranging: If the unknown is in the denominator, multiply both sides by the unknown first, then divide as necessary.

Finding Missing Angles

Key Steps

1. Label the sides: Opposite, Adjacent, Hypotenuse.
2. Select the appropriate equation: Use sine, cosine, or tangent based on the given sides.
3. Use inverse trigonometric functions: To find the angle, use inverse sin, cos, or tan.

Example Problems

Problem 1

• Known: Opposite (5), Adjacent (12)
• Unknown: Angle (X)
• Equation: \tan(X) = \frac{5}{12}
• Solution: X = \text{tan}^{-1}(\frac{5}{12}) ≈ 22.6^{\circ}

Problem 2

• Known: Adjacent (4), Hypotenuse (13)
• Unknown: Angle (X)
• Equation: \cos(X) = \frac{4}{13}
• Solution: X = \text{cos}^{-1}(\frac{4}{13}) ≈ 72.1^{\circ}

Problem 3

• Known: Opposite (5), Hypotenuse (10)
• Unknown: Angle (X)
• Equation: \sin(X) = \frac{5}{10}
• Solution: X = \text{sin}^{-1}(\frac{5}{10}) = 30^{\circ}

Key Concepts

• Labels: Correctly identify hypotenuse, opposite, and adjacent sides.
• Equations: Choose the correct SOH CAH TOA equation based on the sides involved.
• Inverse Functions: Use inverse trigonometric functions to find angles from side lengths.

Summary

• SOH CAH TOA: Used for solving trigonometric problems involving right triangles.
• Steps: Identify sides, use the correct equation, rearrange, and solve.
• Applications: Can find both missing sides and angles using appropriate trigonometric ratios and inverse functions.