Finding sides and angles.

Step 1 of 5

A ladder against a wall

When you know one angle and one side of a right triangle, you can work out any other side. Pick the ratio that links the side you know with the side you want, write it as an equation, and solve it.

A ladder 10 m long leans against an upright wall at $60^\circ$⁠ to the ground. The ladder is the hypotenuse $CA$⁠, the wall is leg $BC$⁠ and the ground is $AB$⁠. Answer, then choose Worked out under Look inside.

Step 2 of 5

A ramp

Again the hypotenuse is known, but now you want the side adjacent to the angle: the ratio that links them is the cosine. A ramp 12 m long rises at $20^\circ$⁠. How much ground does it take up?

Step 3 of 5

The height of a tree

Sometimes the hypotenuse is not known at all. You stand 15 m from a tree and see its top at $35^\circ$⁠ above the ground. The distance is the side adjacent to the angle and the height is the side opposite it: the ratio that links them is the tangent. The bottom line on the board shows its value.

Step 4 of 5

From a ratio to an angle

So far the angle was known. What if two sides are known and the angle is not? Here Side AB is locked at $4$⁠. Find the angle at which the side opposite $A$⁠ comes out as $3$⁠: move Angle A until the tangent is $\frac{3}{4} = 0.75$⁠.

Step 5 of 5

The inverse function

A calculator does the search you just did in one go. The inverse functions, $\sin^{-1}$⁠, $\cos^{-1}$⁠ and $\tan^{-1}$⁠, take a ratio and give back an angle: $\tan^{-1} 0.75 \approx 36.9^\circ$$\tan^{-1} 0.75 \approx$$36.9^\circ$⁠. Here angle $A$⁠ is hidden, and the sides are known. Answer, then choose Worked out under Look inside.