How to use trigonometry in an isosceles triangle?

Splitting the Triangle to Create Right Angles

Basic trigonometric ratios like sine, cosine, and tangent only apply directly to right triangles. To work with an isosceles triangle using these tools, your first move is to draw an altitude from the apex straight down to the base at a 90-degree angle.

Because an isosceles triangle is symmetrical, this altitude acts as an axis of symmetry. It cuts the base into two equal segments and bisects the apex angle. This creates two identical right triangles back to back, allowing you to use right-triangle trigonometry on either side.

Step-by-Step Calculation

First, draw the altitude from the angle between the two equal legs down to the base. Label the altitude h, half the base as b/2, and the equal legs as hypotenuses c. Also note that the top angle is now divided into two equal angles of A/2.

Next, select the trigonometric ratio matching your given measurements. For instance, if you know the equal legs are 10 cm and the base is 12 cm, half the base is 6 cm. In the right triangle, cosine of the base angle equals adjacent over hypotenuse, which is 6/10 or 0.6. Using inverse cosine gives a base angle of approximately 53.13 degrees.

Finally, use the height and base to calculate area if needed. With the Pythagorean theorem or sine, the height is 8 cm. The area of the entire isosceles triangle is half the full base times height: 0.5 * 12 * 8 = 48 square centimeters.

Using the Same Trick in Rectangles and Rhombuses

You can apply this same approach to other symmetric shapes. A rectangle's diagonal divides it into two right triangles where the side lengths are the legs and the diagonal is the hypotenuse. The tangent of the angle between the diagonal and the base is simply the height divided by the width.

In a rhombus, the two diagonals are perpendicular bisectors of each other. Drawing both diagonals divides the rhombus into four congruent right triangles. You can find any side length, perimeter, or interior angle by solving just one of those smaller right triangles.