Geometry Day 62

1. Geometry Day 62 Trigonometry III: The Sine of the Times

2. Today’s Objective • The Law of Sines • The Law of Cosines

3. Review • The trigonometric functions – sine, cosine, and tangent – give the ratios of certain sides given an acute angle of a right triangle. • Remember, you take the sin, cos, and tan of angles, and you will get a ratio.

4. Review • The inverse trig functions will take a ratio and provide the angle that forms that ratio. • In other words, it allows you to solve for an angle given two sides of a right triangle.

5. Trig and non-right triangles • Sine, cosine, and tangent only work on right triangles, but there is a way we can expand this idea to other types of triangles. • Take any triangle. • Let’s say we know two angles anda side opposite one of the knownangles… • …and we want to know the sideacross from the other angle. B x 15 57 21 A C

6. Trig and non-right triangles • If we are to use trigonometry, we need right triangles. Can we create right triangles in this diagram? • Any triangle can be divided into tworight triangles by drawing its altitude. • Can you use the information in the diagram to solve for x? • Hint: solve for h first. B x 15 h 57 21 A C

7. Let’s generalize the process • Can you use sine to find a relationship between the angles of a triangle and their opposite sides? B c a h A C

8. The Law of Sines • This idea becomes what is known as the Law of Sines: • You can use the Law of Sines if you know ASA, AAS, or SSA*. (In other words, if you can compare an angle to the side across from it.) • *SSA can be tricky, in that there may be one, none, or two triangles that exist for a given set of values. All of the problems in this class will only have one solution; you’ll learn more in Pre-Calculus. B c a A C b

9. Practice • Solve for the variables: x

10. The Law of Cosines • If we know SAS or SSS, then we cannot use the Law of Sines, since we can’t form a ratio between an angle and its opposite side. • Let’s examine: • We’ll create right triangles by drawing the altitude. • We’ll give CD a length of y. • What is AD? • If we know what y is, we could use the Pythagorean Theorem to find BD, then use it again to find x. • We can use cosine to find y. B x 7 y 4 – y 62 A C D 4

11. Let’s generalize the process • Let’s label the other segments. • We can use the Pythagorean Theorem twice: B c a h x b – x A C b

12. Let’s generalize the process • We can use cosine to write an expression for x: B c a h x b – x A C b

13. The Law of Cosines • The Law of Cosines states: • Remember, the Law of Cosines can be used if you know SAS or SSS. B c a A C b

14. Practice • Solve for the variables: a

15. Summary • Solving a triangle means to find the measures of all sides and angles. • For a right triangle: • If you know two sides, you can find the third by the Pythagorean Theorem. You can find the acute angles by using inverse trig functions. • If you know a side and acute angle, you can find the other sides with trig functions. You can find the other acute angle with the Triangle Interior Angle Sum Theorem. • For non-right triangles: • If you know ASA or AAS, you can use the Interior Angle Theorem to find the third angle, and the Law of Sines to find the other sides. • If you know SSA, you can use the Law of Sines to find one of the angles. Then you’ll have either ASA or AAS, and see above. • If you know SAS, you can use the Law of Cosines to find the third side. Then you’ll have SSA (see above) and SSS (see below). • If you know SSS, you can use the Law of Cosines to solve for an angle. Then you’ll have SSA (see above).

16. Homework 37 • Workbook, p. 107