Sine, cosine and tangent.

Step 1 of 5

Naming the sides

In the geometry Unit you saw that in a right triangle the side opposite the right angle is called the hypotenuse, and that it is the longest side. Trigonometry names the two legs as well, by the angle you look from. From angle $A$⁠, leg $BC$⁠ is opposite it, and leg $AB$⁠ touches it, adjacent to it. On the board the side opposite $A$⁠ is pink, the side adjacent to $A$⁠ is mint and the hypotenuse is violet. Answer: what changes when you look from angle $C$⁠?

Step 2 of 5

Same angle, same ratio

You saw this in the last Unit too: triangles with the same angles are similar, and all their sides are multiplied by the same number. So the ratio of two sides does not depend on the size of the triangle, only on its angles. The ratio of the side opposite an angle to the hypotenuse is called the sine of the angle:

$\sin A = \frac{BC}{CA}$$\sin A =$$\frac{BC}{CA}$⁠

The bottom line on the board works it out. Angle A is locked at $30^\circ$⁠. Predict what happens to the sine when side $AB$⁠ doubles, then move Side AB yourself.

Step 3 of 5

Working out a sine

Here angle $A$⁠ is $40^\circ$⁠, and its sine is hidden. The lengths of the sides are on the board: divide the side opposite $A$⁠ by the hypotenuse. Once you have answered, choose Worked out under Look inside to see the result.

Step 4 of 5

Cosine

The cosine of the angle is the ratio of the side adjacent to it to the hypotenuse:

$\cos A = \frac{AB}{CA}$$\cos A =$$\frac{AB}{CA}$⁠

As angle $A$⁠ grows, the triangle stands up: the side adjacent to $A$⁠ gets shorter compared with the hypotenuse, and the cosine shrinks. Side AB is locked; move Angle A and watch the bottom line.

Step 5 of 5

Tangent

The third ratio does not use the hypotenuse at all. The tangent of the angle is the ratio of the side opposite it to the side adjacent to it:

$\tan A = \frac{BC}{AB}$$\tan A =$$\frac{BC}{AB}$⁠

It measures how steep the triangle is: how far you rise for each step forward. Many people remember the three as SOH CAH TOA: sine is opposite over hypotenuse, cosine adjacent over hypotenuse, tangent opposite over adjacent. Here $A = 30^\circ$⁠. Predict the tangent when $A$⁠ is $45^\circ$⁠.