5/6/14 Obj: SWBAT apply properties of the Law of Sines and Cosines

Bell Ringer: HW Requests: 13.3 WS

In Class: 13.4 WS

Homework: Complete 13.4 WS

Education is Power!

No Half SteppingJust do the work!

Law of Sines Find missing parts of any triangle - Works for

ANY triangle Not just a right triangle.http://www.youtube.com/watch?v=ETqe-_plC3Y

Solve a triangle using Law of Sines when:1. The measures of 2 ’s and any side are known

(AAS or ASA)2. The measures of 2 sides & an opposite one of the

sides is known. (SSA) (ambiguous case)

Question: In a triangle, can the sine of an angle be negative, Can the cosine be negative?

http://www.youtube.com/watch?v=ETqe-_plC3Y

Law of Sines

Law of Cosines Find missing parts of any triangle - Works for ANY triangleNot just a right triangle.

Solve a triangle using Law of Cosines when:1. There are two sides and an included angle, SAS.2. There are three sides SSSIn class: pg 494 #6, 10, 18, 38 Exit Ticket: pg 494 #8, 12,

20HW:; pg 493 #1-19 odds, 35

Locate the fire

Find the area of the figure

http://www.youtube.com/watch?v=Tqn6-4W39MI&feature=related

Quadrantal Angles

Angles have terminal side and lie on a coordinate axis

http://www.mathsisfun.com/geometry/unit-circle.html

http://www.youtube.com/watch?v=6Qv_bPlQS8E&feature=relmfu

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The values of the trig functions for non-acute angles (Quads II, III, IV) can be found using the values of the corresponding reference angles.

Reference Angles

Definition of Reference Angle

Let be an angle in standard position. Its reference angle is the acute angle formed by the terminal side of and the horizontal axis.

ref

Find the reference angle

• Important idea: the triangle is always drawn to the x axis

• The reference angle is the POSITIVE ACUTE angle made in the triangle closest to the origin

x

y

11’

2

2’

Find the reference angle

• Find the reference angle for 288• Draw 288 angle• Draw triangle back to x axis• Find reference angle x

y

’

• ’ is what is left over from 360• 360 – 288 = ’ • = 72

Find the reference angle

• Find the reference angle for 98• Draw 98 angle• Draw triangle back to x axis• Find reference angle x

y

’

• ’ is what is left over from 180• 180 – 98 = ’ • = 82

Find the reference angle

• Find the reference angle for -55• Draw -55 angle• Draw triangle back to x axis• Find reference angle x

y

’

• ’ is what is the same as measured from zero but the positive angle

• |-55| = ’ • = 55

23

Example

Find the reference angle for 225

Solution y

x

ref

By sketching in standard position, we see that it is a 3rd quadrant angle. To find , you would subtract 180° from 225 °.

ref

225 180

45

ref

ref

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So what’s so great about reference angles?

Well…to find the value of the trig function of any non-acute angle, we just need to find the trig function of the reference angle and then determine whether it is positive or negative, depending upon the quadrant in which the angle lies.

For example,

1sin 225 (sin 45 )

2

45° is the ref angle

In Quad 3, sin is negative

25

Example

Give the exact value of the trig function (without using a calculator).

Express the given trigonometric function in terms of the same function of a positive acute angle.

cos 150

6) tan 91 , sec 345

8) cos 190 , cot 290

http://www.mathsisfun.com/geometry/unit-circle.html

27

Examples (Text p 239 #6 & 8)

Express the given trigonometric function in terms of the same function of a positive acute angle.

6) tan 91 , sec 345 8) cos 190 , cot 290