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Both Acute Angles

Part of the Right-Triangle Trigonometry section of Coddy's Trigonometry journey. Lesson 27 of 50.

Once one acute angle is known, the other is free: the two share the 90° the right angle leaves, so the second is 90° minus the first. In a 3, 4, 5 triangle the angle opposite the 3 is arctan(3/4), about 36.87°, so the angle opposite the 4 is about 53.13°.

$Right triangle with sides 3, 4 and 5, about 36.87° at A and θ at B$

You can also find the second angle directly, reading the sides from it: the 4 is opposite θ and the 3 is adjacent, so theta = atan(4/3), again about 53.13°. Going direct does not lean on the first answer, so a slip there cannot carry over, and the two routes check each other.

Challenge

Triangle ABC has a right angle at C. This time θ is at B: the side opposite θ is 8 and the side next to it is 15. Find θ in degrees, rounded to two decimal places. Then press Check.

Try it yourself

$Use the figure to find theta$

Use the figure to find theta

Test yourself

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

All lessons in Right-Triangle Trigonometry

1Sides of a Right Triangle

How Tall Is the Tree?The Hypotenuse The Opposite Side The Adjacent Side When the Angle Moves Recap: Naming the Sides

4Finding the Hypotenuse

The Unknown Below the Bar How Long Is the Kite String?A Leg Below the Bar Is the Answer Sensible?Recap: The Hypotenuse

7Elevation and Depression

Angle of Elevation Measuring From Eye Level Angle of Depression The Sun's Angle Two Triangles, One Height Recap: Looking Up and Down

2The Three Ratios

The Sine Ratio The Cosine Ratio The Tangent Ratio Choosing the Ratio Recap: The Three Ratios

5Finding an Angle

Inverse Sine Inverse Cosine Inverse Tangent The Angle of a RampBoth Acute AnglesRecap: Finding an Angle

8Reciprocal Ratios

Cosecant, Secant, Cotangent Exact Reciprocal Values A Side From a Cosecant A Ladder and a Secant Recap: Reciprocal Ratios

3Finding a Side

Sides at Any Angle Rounding the Answer Tangent: Leg From Leg A Ladder Against a Wall The Other Acute Angle Recap: Finding a Side

6Special Triangles, Exactly

The Two Special Triangles An Exact Hypotenuse The Side Next to 30°An Exact Angle Recap: Exact Values

9Final Challenges

Is the Ladder Safe?The Wheelchair Ramp Two Observers The Landing Plane The Kite The Sun's Angle, Exactly