Finding Areas with Trigonometry

Objectives I can use trigonometry to find the area of a triangle.

A.A B.B C.C D.D A.346 m 2 B m 2 C.173 m 2 D m 2 Find the area of a regular triangle with a side length of 18.6 meters. Practice

Next Application… Area of an oblique triangle – Given two sides of any triangle and the measure of an angle between them – Use trigonometry to find its surface area Recall previous formula for the area of a triangle: A = ½ bh

We will use an obtuse triangle Label sides a, b, and c, opposite their corresponding angles Draw a height, h, inside

Next… In order to use A = ½ bh, we need b and h, but all we know are a, b, and the measure of angle C (for example) we need “h”! Look at triangle BDC inside: – How can we write a trig ratio using sides h and a? – We can use this to solve for “h”!

So Far we have… Solve this for “h”: h = a sin C Now we have the info we need to use A = 1/2bh! A = ½ bh substitute “a sin C” for “h” A = ½ a b sin C

IN CONCLUSION The area of an oblique triangle is one-half the product of the lengths of two sides, times the sine of their included angle! For any triangle, ABC Area = ½ bc sinA = ½ ab sinC = ½ ac sinB

Practice Find the area of a triangular lot having two sides of lengths 90m and 52m and an included angle of 102°. Draw it: Area = ½ (90)(52) sin 102 ≈ m 2

Practice Find the area of a triangle with sides 6 and 10 and an included angle of 110° Round to the nearest hundredth. Area = 28.19

Practice Find the area of a triangle with side lengths 92 and 30 with an included angle 130°. Area =