Warm Up = = 9 5 3) = = ) ) 99 = ) Simplify. = = 4 6 = 2 6 3

Transcription

1 Warm Up Simplify. 1) 99 = ) = = 9 5 3) = ) = = = = 2 6 3

2 Test Results Average Median 5 th : th : th :

3 8-1 Simple Trigonometric Equations Objective: To Solve Simple Trigonometric Equations and Apply Them

4 180 Name 4 angles between 0 and 360 that have a reference angle of , is called the referenc e for 150, 210, & 330

5 Reference Angles If you know the reference angle, use these formulas to find the other quadrant angles that have the Degrees 180 π Radians same reference angle = reference angle π + 2π

6 There are many solutions to the trigonometric equation sin x = 1 2 y -19π 6-11π 6-7π π 5π 13π π π 6-3π -2π -π π 2π 3π 4π x y = 2 1 All the solutions for x can be expressed in the form of a general solution. Copyright by Houghton Mifflin Company, Inc. All rights reserved. 6

7 We know that x = π 6 and x = 5π 6 are two solutions. Since the period of sin x is 2π, we can add integral multiples of 2π to get the other solutions: x = π 6 + 2nπ and x = 5π 6 + 2nπ, n is any integer.

8 Solving for angles that are not on the Unit Circle Example 1: Find the values of 0 < x < 360 for which sinx = 0.35 y 1-1 x degrees

9 Solving for angles that are not on the Unit Circle Example 1: Find the values of 0 < x < 360 for which sinx = 0.35 Step 1 Find the inverse on the calculator. Make sure you are in the right mode. Use the inverse key. x 20.5

10 Example 1: Find the values of 0 < x < 360 for which sinx = 0.35 Step 2 Find the reference angle. Since the answer given by your calculator is NOT between 0 and 360 degrees, find the proper answers by using RA. RA for 20.5 is 20.5

11 Example 1: Find the values of 0 < x < 360 for which sinx = 0.35 Step 3 Determine which quadrant(s) the answer(s) is(are) in. Find the answers using the RA formulas. Sine is negative in QIII & QIV, so use and 360 to find the angles.

12 If you had been asked to find ALL values of x for which sin x = 0.35, then your answer would be: x n AND x n, for any integer n.

13 sinx = 0.35 x and y 1-1 x degrees

14 Solving Graphically

15 Example 2 Find the values of x between 0 and 2π for which sin x = 0.6 Step 1: Set the calculator in radian mode and use the inverse sine key.

16 Step 2 Find the reference angle is in QI so it is the reference angle for other solutions. Step 3: Determine the Proper Quadrant Since sin x is positive, the answers are in Quadrant I and Quadrant II. For Q2: x = π Final answers: x = and

17 If you had been asked to find ALL values of x for which sin x = 0.6, then your answer would be: x πn AND x πn, for any integer n.

18 Example 2: Find the values of x between 0 and 2π for which sin x = 0.6 Method 2: Graphically Step 1 Set the calculator in radian mode.

19 Use your knowledge of trig functions to choose an appropriate window Use the intersect key once more for the second point of intersection.

20 Example 3 To the nearest tenth degree, solve: 3 cos θ + 9 = 7 for 0 o θ 360 o First apply the basic algebra rules and isolate the variable. 3 cos θ + 9 = 7 3 cos θ = 2 cos θ = 2 3

21 Find the appropriate quadrant Since cos θ < 0, the final answers are in QII and QIII. Use your knowledge of reference angles to find the second answer. The RA for is 48.2 The QIII angle is The final answers are: θ or θ 228.2

22 Example 3 Graphing Calculator:

23 Solving equations with secθ, cscθ, and cotθ 1. Isolate the trig function. 2. Convert to sine, cosine, or tangent. 3. Solve for the new trig function. Example 4 cscθ = 1 sinθ, so Solve 4cscθ 3 = 9 for 0 θ < 2π 4cscθ = 12 cscθ = 3 1 sinθ = 3 sinθ = 1 3 θ = sin θ = 0.34, so RA = 0.34 Sine is positive in Q1 & Q2 θ = and θ = π 0.34 θ = 2. 80

24 Practice 1. Solve cosθ = for 0 θ , Solve cscθ = 14 for 0 θ Solve 5cotθ 3 = , for 0 θ 2π 1.35, 4.49

25 Classwork

26 Homework: A:Page 299 #1-21 odds, 31,33,35 B: Page 229#1,5,9,13-21 odds, 31,33,35,45,46