8-1 Simple Trigonometric Equations. Objective: To solve simple Trigonometric Equations and apply them

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1 Warm Up Use your knowledge of UC to find at least one value for q. 1) sin θ = 1 2 2) cos θ = 3 2 3) tan θ = 1 State as many angles as you can that are referenced by each: 1) 30 2) π 3 3) 0.65 radians Useful information to MEMORIZE: π π π 6.28

2 8-1 Simple Trigonometric Equations Objective: To solve simple Trigonometric Equations and apply them

3 There are many solutions to the trigonometric equation sin x = π 6-11π 6-7π 6 y π π 6 13π 6 17π 6 25π 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. 3

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

5 Solving for angles that are not on UC We will work through solutions algebraically and graphically. Learning both methods will enhance your understanding of the work.

6 y degrees x -1

7 Example 1: Find the values of 0 < x < 360 for which sinx = 0.35 Method 1: Algebraically: Step 1 Set the calculator in degree mode and use the inverse sine key x 20.5

8 Find the final answer(s) for the given range. Since the answer given by your calculator is NOT between 0 and 360 degrees, find the proper answers by using RA.

9 y degrees x -1

10 Solving Graphically

11 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.

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

13 Step 2: Determine the proper quadrant is the reference angle for other solutions. Since sin x is positive, a Quadrant II angle also satisfies the equation.

14 x = π Final answers are: x and

15 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.

16 Example 1: 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.

17 Use your Knowledge of trig functions to choose an appropriate window Use the intersect Key once more for the second point of intersection. i.e solution.

18 When you use the graphing method, you can easily see there is more than one solution. When using the graphing method, it might take a while to set the window properly. The algebraic method is quicker, however, you have make sure to look for a possible second answer.

19 To solve an equation involving a single trigonometric function, we first transform the equation so that the function is alone on one side of the equals sign. Then we follow the same procedure used in Example 1.

20 Example 2 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 the QII and QIII. Use your knowledge of reference angle to find the second answer: The final answers are: θ or θ 228.2

22 Another way; ignore the negative sign. The reference angle is: cos 1 ( 2 3 ) 48.2o The first solution is: θ 180 o 48.2 o = o The second solution is: θ 180 o o = o

23 Graphing Calculator: Although this is a reasonable window to start with, it does not capture the graph. So change Ymin and Ymax.

24

25 Warm Up Day 2; 1. cosθ = 2 a) has 0 solution. b) has 1 solution. c) has 2 solutions. d) has infinite number of solutions. 2. tanθ = 2 a) has 0 solution. b) has 1 solution. c) has 2 solutions. d) has infinite number of solutions. Graph sine, cosine and tangent functions.

26 Inclination and Slope The inclination of a line is the angle a, where 0 o a < 180 o, that is measured from the positive x-axis to the line.

27 Inclination and Slope The inclination of a line is the angle a, where 0 o a < 180 o, that is measured from the positive x-axis to the line. The line at the left below has inclination 35 o. The line at the right below has inclination 155 o. The theorem that follows states that the slop of a nonvertical line is the tangent of its inclination.

28 Theorem For any line with slope m and inclination a m = tan a if a 90 o. If a = 90 o, than the line has no slope. (The line is vertical.)

29 Example 3 to the nearest degree, find the inclination of the line 2x + 5y = 15 Solution: rewrite the equation as y = 2 5 x + 3 Slope = = tan a a = tan o ( the reference angle is 21.8 o.)

30 Since tan a is negative and a is positive angle, 90 o < a < 180 o the inclination is 180 o 21.8 o o. In section 6-7, you learned to graph conic sections whose equations have no xy term. That is equation of the form.

31 Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0 Where B=0. the graph at the right shows conic section with center at the origin whose equation has an xy term (B 0). Conics like this have one of their two axes inclined at an angle a to the x axis. To find this direction angle a, use the formula below.

32 a = π 4 if A=C Tan 2a = B A C if A C, and 0< 2a < π The direction angle a is useful in finding the equation of the axes of these conic sections. This is shown in method 1 of example 4 on the next page

33 Homework: Sec 8.1 Written exercises #1-21 odds Optional: Sec 8.2 written exercises ALL

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