Lesson 2.9

Evaluating Trigonometric Ratios

Combine everything you know. Use the Pythagorean Theorem to find a missing side, then construct all six trigonometric functions for a given angle.

Introduction

Up until this point, we have been handing you right triangles that already have all three sides perfectly calculated. In the real world, you are almost never given all three sides! Your goal today is to synthesize the Pythagorean Theorem (from Unit 1) with your library of six trigonometric functions to completely "solve" a triangle from scratch.

Past Knowledge

You know how to use to find a missing side, and you know how to build the trig fractions.

Today's Goal

Given a triangle with only two sides, mathematically calculate the third, and then output all six primary and reciprocal trig ratios.

Future Success

This four-step evaluation process is the foundation for almost every standardized test trigonometry question you will face.

Key Concepts

The Master Workflow

To evaluate a full set of trigonometric ratios from an incomplete triangle, follow this exact sequence every single time to avoid chaos:

1

Find the Missing Side

Use the Pythagorean Theorem (). Remember that must always be the Hypotenuse (the side across from the angle). If a leg is missing, you must subtract!

2

Label the Roles

Stand at the target reference angle given in the problem. Clearly label the Opposite, Adjacent, and Hypotenuse sides.

3

Write SOH CAH TOA

Construct the primary fractions for Sine, Cosine, and Tangent using the three side lengths you've gathered.

4

Flip for Reciprocals

Take the three primary fractions and turn them upside down to instantly generate Cosecant, Secant, and Cotangent. Rationalize square roots if they end up in the denominator.

Visualization

Consider a triangle where we only know the two legs are and . Watch how the missing hypotenuse unlocks all the remaining ratios.

Step 1: Pythagoras
Primary
Reciprocal

Worked Examples

Basic

Missing Hypotenuse

Question: The two legs of a right triangle are and . If angle is across from the -unit leg, list the values of and .

Step 1: Find the Hypotenuse

Step 2: Label Roles for Angle

  • Opposite to is (given in problem description).
  • Adjacent to is the other leg, .
  • Hypotenuse is .

Step 3: Evaluate Cosine (CAH)

Step 4: Evaluate Cotangent (Flipped TOA)

Tangent is . Therefore, Cotangent is !

Final Answer: and

Intermediate

Missing Leg Constraint

Question: In a right triangle, the hypotenuse is and the adjacent side to angle is . Find .

Step 1: Setup Pythagoras to find the missing leg ()

Warning! The 25 must be the variable.

Because the side adjacent was , the side we just found () is the Opposite side.

Step 2: Find Cosecant

Cosecant is the reciprocal of Sine. Sine is . Therefore, Cosecant is .

Final Answer:

Advanced

Variables and Radicals

Question: You are told that . Using this information, construct the reference right triangle and calculate the exact value of . Rationalize all denominators.

Step 1: Reverse-engineer the triangle from Tangent.

We know Tangent is TOA (Opposite over Adjacent). So, if , then the Opposite leg is inherently , and the Adjacent leg is inherently .

Step 2: Find the Hypotenuse

Step 3: Calculate Cosine

Cosine is CAH (Adjacent over Hypotenuse).

Step 4: Rationalize

Final Answer:

Common Pitfalls

Blindly Adding the Squares

When applying the Pythagorean theorem, the most disastrous mistake is automatically squaring the two given numbers and adding them together, completely ignoring which sides of the triangle they represent.

❌ Incorrect: You are given a leg of 4 and a hypotenuse of 5. You calculate . (You just made the hypotenuse longer than the hypotenuse!)

✅ Correct: Stop and write the formula every time! .

Real-Life Applications

Video Game Physics Engines

When a character in a video game points a weapon at a target, the game engine only knows two raw numbers: the X-coordinate difference and the Y-coordinate difference between the player and the enemy. To calculate the 3D rotating physics of the character's arm, the engine must instantaneously run the Pythagorean Theorem to find the direct distance to the target, and then evaluate the ArcTangent and Cosine ratios of those side lengths to properly render the character's skeletal joint angles. This full evaluation loop happens times per second!

Practice Quiz

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