Trigonometric Ratios: Solving Right Triangles

Lesson 2 of 2: From Definition to Application

In this lesson:

• Use trig ratios to find unknown sides (given angle + one side)
• Use inverse trig to find unknown angles (given two sides)
• Apply to real-world problems

What You Will Learn Today

By the end of this lesson, you should be able to:

1. Use sine, cosine, and tangent to find unknown side lengths in right triangles
2. Use inverse trigonometric functions (sin⁻¹, cos⁻¹, tan⁻¹) to find unknown angle measures

Can We Find the Height Without Measuring?

A ladder leans against a wall at a 70° angle.
The ladder is 12 feet long.

Can we find how high up the wall it reaches — without measuring?

We know the angle and the hypotenuse. We want the opposite side.

Which ratio connects opposite and hypotenuse?

A Four-Step Procedure for Any Problem

To find an unknown side in a right triangle:

Step 1: Identify the given angle and the given side
Step 2: Identify the unknown side (opposite? adjacent? hypotenuse?)
Step 3: Choose the ratio that connects all three: given angle, given side, unknown side
Step 4: Write the equation and solve algebraically

SOH-CAH-TOA tells you which ratio to use based on the sides involved.

Decision Tool: Identifying the Right Ratio

Identify your sides first — the ratio follows automatically.

Example A: Find the Opposite Side

Right triangle: angle = 40°, hypotenuse = 15. Find the opposite side.

Sides involved: opposite (unknown) and hypotenuse (given) → use sine

Example B: Unknown in the Denominator

Right triangle: angle = 55°, opposite = 20. Find the hypotenuse.

Sides involved: opposite (given) and hypotenuse (unknown) → use sine

Quick Check: Set Up the Equation

Right triangle: angle = 30°, adjacent = 10. Find the opposite side.

Which ratio? Set up the equation — don't solve yet.

(Adjacent and opposite → tangent. Answer: )

Example C: Find Opposite Side Using Tangent

Right triangle: angle = 30°, adjacent = 10. Find the opposite side.

Adjacent and opposite → use tangent

Your Turn: Find the Adjacent Side

Right triangle: angle = 50°, hypotenuse = 20. Find the adjacent side.

Step 1: Sides involved — adjacent (unknown) and hypotenuse (given)
Step 2: Which ratio? (Adjacent + hypotenuse → cosine)

Step 3: Solve: adjacent = 20 × cos(50°) ≈ 20 × ?

Use a calculator to find cos(50°), then complete the calculation.

When the Unknown Is in the Denominator

Pattern: When the unknown side is in the denominator:

• Cross-multiply, or equivalently divide the known value by the trig ratio
• Never multiply — that gives you the wrong answer

Real-World Example: The Ladder Problem

The ladder (12 ft) makes a 70° angle with the ground. How high up the wall?

Practice: Find the Unknown Side

Identify which ratio to use, then find the unknown side.

1. Angle = 40°, hyp = 25, find opposite
2. Angle = 65°, adj = 8, find hypotenuse
3. Angle = 45°, opp = 10, find adjacent

Work each one before the answer slide.

Practice: Check Your Answers Here

Problem 1 (sine):

Problem 2 (cosine, denominator):

Problem 3 (tangent — tan(45°) = 1 exactly):

Transition: Working Backwards to Find Angles

So far: Known angle + known side → find unknown side

New direction: Known sides → find the angle

"If sin(θ) = 0.5, what is θ?"

This requires inverse trigonometric functions — the undoing of sine, cosine, and tangent.

Inverse Trig: Undoing the Function

Forward: Input angle → output ratio

Backward: Input ratio → output angle

Notation: is also written arcsin. Similarly (arccos) and (arctan).

Read as: "the angle whose sine is 0.5"

Using the Calculator for Inverse Trig

To find , , or on a calculator:

1. Press 2nd or Shift key
2. Then press sin, cos, or tan
3. Enter the ratio value and press enter

Example: Find the angle if sin(θ) = 0.625

• Press: 2nd → sin → 0.625 → Enter
• Result: 38.7°

Example D: Find the Angle (Using sin⁻¹)

Right triangle: opposite = 5, hypotenuse = 8. Find angle θ.

Sides involved: opposite and hypotenuse → use sine

Example E: Find the Angle with Cosine Inverse

Right triangle: adjacent = 7, hypotenuse = 10. Find angle θ.

Sides involved: adjacent and hypotenuse → use cosine

Quick Check: Which Inverse Function?

Right triangle: opposite = 9, adjacent = 12.

Which inverse function do you need to find angle θ?

Set up the ratio and identify the function:

(Opposite and adjacent → tangent → → )

Example F: Find the Angle with Tangent Inverse

Right triangle: opposite = 9, adjacent = 12. Find angle θ.

Sides involved: opposite and adjacent → use tangent

Real-World Application: The Ramp Angle

A ramp rises 3 ft over a horizontal distance of 15 ft. What angle does it make with the ground?

Practice: Find the Missing Angle

Find angle θ in each right triangle.

1. Opposite = 6, hypotenuse = 10 → compute ratio, then apply
2. Adjacent = 9, hypotenuse = 15 → compute ratio, then apply

Show your ratio computation and your inverse trig setup before checking.

Practice: Check Your Angle Answers Here

Problem 1: Opposite = 6, hypotenuse = 10 → sine

Problem 2: Adjacent = 9, hypotenuse = 15 → cosine

Notice: both gave ratio 0.6, but different inverse functions → different angles!

Key Takeaways: What You Now Know

✓ Finding sides: Identify angle + given side → pick ratio → write equation → solve

✓ Finding angles: Identify two sides → pick ratio → compute decimal → apply inverse

✓ , , are inverse functions — they give the angle from the ratio

Watch out: , NOT — always apply the inverse

Watch out: These formulas require a right angle — don't use them for non-right triangles

Watch out: Compute the ratio first, then apply the inverse function — don't skip steps

What Comes Next in This Unit

HSG.SRT.C.7: Relationships between trig ratios for complementary angles

• You discovered it on the last slide: sin(36.9°) = cos(53.1°) = 0.6
• Complementary angles sum to 90° — and their sine and cosine swap

HSG.SRT.C.8: Solving right triangles completely

• Given enough information, find ALL unknown sides and angles
• Combines Pythagorean Theorem with trig ratios

The same three ratios — deeper connections ahead.