The Side Next to 30°

Part of the Right-Triangle Trigonometry section of Coddy's Trigonometry journey. Lesson 31 of 50.

In the 30-60-90 triangle the middle side, the one next to the 30° angle, carries the √3. With a hypotenuse of 10, the side next to 30° is 10×√(3)/2.

On the board: the side touches 30°, so it is adjacent, and cos(30°) = y/10. cos 30° is √3/2, so y = 5√(3), about 8.66. The short side, opposite 30°, is 5. Check with Pythagoras: 5² + (5√3)² = 25 + 75 = 100 = 10².

$Right triangle with a 30° angle, hypotenuse 10, opposite side 5 and adjacent side y$

Challenge

Triangle PQR has a right angle at P and a 30° angle at Q. The hypotenuse is 12. Find the side x next to the 30° angle exactly. Then press Check.

Try it yourself

$Use the figure to find x$

Use the figure to find x

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 Ramp Both Acute Angles Recap: 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 HypotenuseThe 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