Name: ______________________ Date: ___________________

Foundations of Mathematics 11

Chapter 4- Oblique Triangle Trigonometry 4.2: Proving and Applying the Sine and Cosine Laws for Obtuse Triangles

We have already shown that the sine law works for acute triangles. Now we are going to try to prove the

sine law for obtuse triangles. Follow the steps of the investigation to prove that the sine law also applies

to obtuse triangles.

A. Draw an obtuse triangle ABC with height AD.

B. Write equations for sin (180o – ) and sin C using the two right triangles.

C. Use the transitive property to make the two expressions for AD equal to each other, and then create

a ratio.

D. Draw a new height, h, from B to base b in the triangle.

E. Write equations for sin A and sin C using the two right triangles.

F. Use the transitive property to make the two expressions for h equal to each other.

Example 1 – In an obtuse triangle, B measures 23.0

o and its opposite side, b, has a length of 40.0 cm.

Side a is the longest side of the triangle, with a length of 65.0 cm. Determine the measure

of to the nearest tenth of a degree.

In lesson 3.3 we proved the cosine law for acute triangles. Follow the steps of the investigation to prove

that the cosine law also applies to obtuse triangles.

A. Use as shown below.

B. Extend the base of the triangle to D, creating two overlapping right triangles, and ,

with height BD. Note on your diagram that two angles are formed at C, and , such that

180o – .

C. Let side CD be x. Use the Pythagorean Theorem to write two expressions for h2, using the two right

triangles.

D. Use the transitive property to make the two expressions for h2 equal to each other. Re-arrange to

isolate c2.

E. In the small right triangle, use a primary trig ratio to write an expression for x.

F. Substitute your expression for x into your equation from part D.

Remember that cos (180° - ACB) = -cos ACB:

c2 = a

2+b

2-2ab cosACB

Example 2 – The roof of a house consists of two slanted sections as

shown. A roofing cap is being made to fit the crown of the roof,

where the two slanted sections meet. Determine the measure of the

angle needed for the roofing cap, to the nearest tenth of a degree.

Key Ideas:

The sine law and cosine law can be used to

determine unknown side lengths and angle

measures in obtuse triangles.

The sine law and cosine law are used with

obtuse triangles in the same way that they

are used with acute triangles.

Be careful when using the sine law to

determine the measure of an angle. The

inverse sine of a ratio always gives an

acute angle, but the supplementary angle

has the same ratio. You must decide

whether the acute angle, , or the obtuse

angle 180o – is the correct angle for your

triangle.

The measures of the angles determined using the cosine laws are always correct.

Assignment: pg. 170 #1, 2abc, 3ab, 4ab, 6, 9, 12