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The Outer Function

Part of the Trig Equations and Inverses section of Coddy's Trigonometry journey. Lesson 12 of 52.

Find $\sin\left(\arccos\left(\frac{3}{5}\right)\right)$ . The inner angle has $\cos\left(\theta\right) = \frac{3}{5}$ : adjacent 3, hypotenuse 5. The missing side is the opposite one, and Pythagoras makes it 4.

Read the outer function off the triangle: sine is opposite over hypotenuse, so $\sin\left(\arccos\left(\frac{3}{5}\right)\right) = \frac{4}{5}$ . Which side was missing depends on the inner function; which ratio you read depends on the outer one. Keep the two jobs apart.

Challenge

Find the exact value of $y = \tan\left(\arccos\left(\frac{4}{5}\right)\right)$ . Draw the triangle on paper, then write y = … as a fraction and press Check.

Try it yourself

y = \tan\left(\arccos\left(\frac{4}{5}\right)\right)

Get $y$ alone

Test yourself

This lesson includes a short quiz. Start the lesson to answer it and track your progress.

All lessons in Trig Equations and Inverses

1Your First Trig Equations

One Equation, Two Angles Reading Signs on the Circle Recap: First Equations

4Compositions

Draw the TriangleThe Outer FunctionA Negative Inside When Undoing Does Not Undo Recap: Compositions

7General Solutions

Every Solution Tangent's Single Family Families in Radians Families, Rounded Recap: Every Solution

10Identities First

Write sin 2x Out Pick the Right cos 2x Swap cos²x for sin²x sec, csc and cot Recap: Identities First

2Inverses and Their Ranges

Why arcsin Gives One Angle The Range of arccos The Range of arctan Recap: Inverse Ranges

5Equations on a Domain

A Negative Cosine Tangent Equations Equations in Radians Two Turns, Four Answers Recap: On a Domain

8Multiple Angles

Two Turns for 2x Solving sin 2x Half Angles Every Solution of sin 3x Recap: Multiple Angles

11Final Challenges

Final: The Bicycle Pedal Final: The Big Wheel Final: Every Solution Final: An Identity and More Final: An Inverse Inside

3Evaluating Inverse Trig

Rounded Angles in Radians An Angle From a Slope Recap: Inverse Values

6Off the Special Angles

Both Answers, Rounded A Negative Sine, Rounded Cosine, Rounded Tangent, Rounded Rounded, in Radians Recap: Off the Table

9Factoring Trig Equations

A Common Factor Never Divide by sin x A Quadratic in Sine A Case With No Angle Square Roots Give Two Signs Recap: Factoring