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An Angle From a Slope

Part of the Trig Equations and Inverses section of Coddy's Trigonometry journey. Lesson 9 of 52.

A road sign says 12%, and most people read it as a 12° slope. It is not.

$A right triangle with a rise of 12 over a run of 100 and the angle θ at its foot$

12% means the road rises 12 m for every 100 m along. So $\tan\left(\theta\right) = \frac{12}{100}$ , and θ is $\arctan\left(0.12\right)$ , about 6.84°. A road going down has a negative grade and a negative angle: arctan(−c) = −arctan(c).

Challenge

A road drops 8 m for every 100 m along. That is a grade of −8%, so $\tan\left(x\right) = -0.08$ . Find its angle x to two decimal places. The angle is negative because the road goes down. Done means x is in degrees with 2 decimals.

Then press Check.

Try it yourself

\tan x = -0.08

Get $x$ alone $-90^{\circ} < x < 90^{\circ}$

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 Triangle The Outer Function A 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 RadiansAn Angle From a SlopeRecap: 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