© 2008 National Council of Teachers of Mathematics http://illuminations.nctm.org

The Law of Sines NAME ___________________________

Right triangle trigonometry can be used to solve problems involving right triangles. However, many interesting problems involve non-right triangles. In this lesson, you will use right triangle trigonometry to develop the Law of Sines. The law of sines is important because it can be used to solve problems involving non-right triangles as well as right triangles.

Consider oblique ǻABC shown to the right. 1. Sketch an altitude from vertex B.

2. Label the altitude k. 3. The altitude creates two right triangles inside 'ABC .

Notice that �A is contained in one of the right triangles, and �C is contained in the other. Using right triangle trigonometry, write two equations, one involving sin A, and one involving sin C.

sin A sin C

4. Notice that each of the equations in Question 3 involves k. (Why does this happen?) Solve each

equation for k. 5. Since both equations in Question 4 are equal to k, they can be set equal to each other. (Why is

this possible?) Set the equations equal to each other to form a new equation. 6. Notice that the equation in Question 5 no longer involves k. (Why not?) Write an equation

equivalent to the equation in Question 5, regrouping a with sin A and c with sin C.

Precalculus: 4.7 Law of Sines/Cosines Part 1 (Day 4) Law of Sines

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

© 2008 National Council of Teachers of Mathematics http://illuminations.nctm.org

The Law of Cosines NAME ___________________________

The law of sines can be used to determine the measures of missing angles and sides of triangles when the measures of two angles and a side (AAS or ASA) or the measures of two sides and a non-included angle (SSA) are known. However, the law of sines cannot be used to determine the measures of missing angles and sides of triangles when the measures of two sides and an included angle (SAS) or the measures of three sides (SSS) are known. Since the law of sines can only be used in certain situations, we need to develop another method to address the other possible cases. This new method is called the Law of Cosines.

To develop the law of cosines, begin with 'ABC . From vertex C, altitude k is drawn and separates side c into segments x and c – x. (Why can the segments be represented in this way?) 1. The altitude separates 'ABC into two right triangles.

Use the Pythagorean theorem to write two equations, one relating k, b, and c – x, and another relating a, k, and x.

2. Notice that both equations contain k2. (Why?) Solve each equation for k2.

!!Law of Cosines !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

!!

Precalculus: 4.7 The Law of Sines/Cosines (Part 1) Day 4 Notes

Right Triangles

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sinθ =opphyp

€

cscθ =hypopp

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cosθ =adjhyp

€

secθ =hypadj

€

tanθ =oppadj

€

cotθ =adjopp

Law of Sines

€

sinAa

=sinBb

=sinCc

Law of Cosines

€

c 2 = a2 + b2 − 2abcosC

SOH$CAH$TOA( Use(this(when(

the(triangle(is(a(right(triangle.(

Law(of(Sin

es( Use(this(when(

the(triangle(is(NOT(a(right(triangle(and(you're(given(an(angle(and(its(opposite(side.(( Law(of(Cosines( Use(this(when(

the(triangle(is(not(a(right(triangle(and(you(don't(have(an(angle(and(its(opposite(side.(

Examples: Solve each of the following triangles. Round all angles to the nearest degree and all sides to the nearest tenth. 1. A = 72o, c = 6, a = 8

2. a = 3, c = 7, B = 130o

3. A = 70o, B = 80o, c = 2

4. a = 5, b = 10, c = 12

5. On May 18, 1980, Mount Saint Helens, a volcano in Washington, erupted with such force that the top of the mountain was blown off. To determine the new height at the summit of Mount Saint Helen, a surveyor measured the angle of elevation to the top of the volcano to be 38o. The surveyor moved 100 feet closer to the volcano and measured the angle of elevation to be 40o30’. Determine the new height of Mount Saint Helens.