warm up. 8-1 simple trigonometric equations objective: to solve simple trigonometric equations and...

Warm Up

• Use your knowledge of UC to find at least one value for q.

• State as many angles as you can that are referenced by each:

1) 0.65 radians

Useful information to MEMORIZE:

8-1 Simple Trigonometric Equations

Objective: To solve simple Trigonometric Equations and apply

them

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3

-1

x

y

1-19π

6-11π

6 -7π

6 π

6 5π

6 13π

6 17π

6 25π

6

-π-2π-3π π 2π 3π 4π

All the solutions for x can be expressed in the form of a general solution.

y =2

1

There are many solutions to the trigonometric equation sin

• We know that and are two particular solutions.

• Since the period of sin is can add integral multiples of to get the other solutions:

• and When is any integer

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.

90 180 270 360

-1

1

x

y

degrees

Example 1: Find the values of for which

Method 1: Algebraically: Step 1 Set the calculator in degree mode and use the inverse sine key

𝑥≈−20.5°

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.

Check your answers:

90 180 270 360

-1

1

x

y

degrees

Solving Graphically

Zoom Trig

Use the intersect key again to find the second value.

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

Example 2: Find the values of between and for which sin

Method 1: Algebraically: Step 1 Set the calculator in radian mode and use the inverse sine key

Step 2: Determine the proper quadrant

is the reference angle for other solutions. Since sin is positive, a Quadrant II angle also satisfies the equation.

Final answers are:

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

Example 1: Find the values of between and for which sin

Method 2: Graphically: Step 1 Set the calculator in radian mode.

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.

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.

• 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.

Example 2

To the nearest tenth degree, solve: cos for

First apply the basic algebra rules and isolate the variable.

Find the appropriate quadrant

Since , the final answers are in the QII and QIII.Use your knowledge of reference angle to find the second answer:

The final answers are:or

Another way; ignore the negative sign.

The reference angle is: The first solution is:

The second solution is:

Graphing Calculator:

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

Warm Up Day 2;

a) has 0 solution.b) has 1 solution.c) has 2 solutions.d) has infinite number of

solutions.

a) has 0 solution.b) has 1 solution.c) has 2 solutions.d) has infinite number of

solutions.

• Graph sine, cosine and tangent functions.

Inclination and Slope • The inclination of a line is the angle , where ,

that is measured from the positive x-axis to the line.

𝟏𝟒𝟑° 𝟑𝟒°

Inclination and Slope

• The inclination of a line is the angle , where , that is measured from the positive x-axis to the line. The line at the left below has inclination The line at the right below has inclination . The theorem that follows states that the slop of a nonvertical line is the tangent of its inclination.

Theorem

• For any line with slope and inclination if .• If , than the line has no slope. (The line is

vertical.)

Example 3to the nearest degree, find the inclination of the line

Solution: rewrite the equation as Slope = -

( the reference angle is .)

• Since tan is negative and is positive angle, the inclination is .

• In section 6-7, you learned to graph conic sections whose equations have no . That is equation of the form.

• Where B=0. the graph at the right shows conic section with center at the origin whose equation has an . Conics like this have one of their two axes inclined at an angle to the x axis. To find this direction angle , use the formula below.

if A=C Tan if AC , and 0The 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

Homework:

• Sec 8.1 Written exercises #1-21 odds• Optional: Sec 8.2 written exercises 22-32 ALL