Warm Up 4) 30° 5) 𝜋 3 6) 0.65 radians sin 𝜃= 1 2 cos 𝜃=− 3 2 tan 𝜃=1 Use your knowledge of the UC to find at least one value for q. sin 𝜃= 1 2 cos 𝜃=− 3 2 tan 𝜃=1 State as many angles as you can that have a reference angle of: 4) 30° 5) 𝜋 3 6) 0.65 radians

Test Results Average Median 2nd: 80.2 80 3rd: 82.2 84.7 4th: 82.8 81.3

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

Reference Angles ∝ 𝜋−∝ ∝ 𝜋 + ∝ 2𝜋−∝ Degrees Radians If you know the reference angle, use these formulas to find the other quadrant angles that have the same reference angle ∝ 180 − ∝ 𝜋−∝ ∝ 180 + ∝ 360−∝ 𝜋 + ∝ 2𝜋−∝

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. There are many solutions to the trigonometric equation sin 𝑥= 1 2 -1 x y 1 -19π 6 -11π -7π π 5π 13π 17π 25π y = -π -2π -3π π 2π 3π 4π All the solutions for x can be expressed in the form of a general solution. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. y=sin x

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.

Example 1: Find the values of 0°<𝑥<360°for which sin𝑥=−0.35 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 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. RA for −20.5° is 20.5° Sine is negative in QIII & QIV, so use 180 + ∝ and 360−∝ to find the angles. Check your answers:

sin𝑥=−0.35 𝑥≈200.5° and 339.5°

If you had been asked to find ALL values of 𝑥 for which sin 𝑥=−0 If you had been asked to find ALL values of 𝑥 for which sin 𝑥=−0.35 , then your answer would be: 𝒙≈𝟐𝟎𝟎.𝟓+𝟑𝟔𝟎𝒏 AND 𝒙≈𝟑𝟑𝟗.𝟓+𝟑𝟔𝟎𝒏, for any integer 𝑛.

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

Step 2: Determine the Proper Quadrant 0.6435 is in QI so it is the reference angle for other solutions. Since sin 𝑥 is positive, the answers are in Quadrant I and Quadrant II. For Q2: 𝑥=𝜋−0.6435≈2.4981. Final answers: 𝒙=𝟎.𝟔𝟒𝟑𝟓 𝒂𝒏𝒅 𝟐.𝟒𝟗𝟖𝟏.

If you had been asked to find ALL values of 𝑥 for which sin 𝑥=0 If you had been asked to find ALL values of 𝑥 for which sin 𝑥=0.6 , then your answer would be: 𝒙≈𝟎.𝟔𝟒𝟑𝟓+𝟐𝝅𝒏 AND 𝒙≈𝟐.𝟒𝟗𝟖𝟏+𝟐𝝅𝒏, for any integer 𝑛.

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

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 131.8° is 48.2° The QIII angle is 180 + ∝ The final answers are: 𝜃≈131.8° or 𝜃≈228.2°

Practice Solve 𝒄𝒐𝒔𝜽=𝟎.𝟒𝟐 for 0°≤𝜃≤360° Solve 𝒄𝒔𝒄𝜽=𝟏𝟒 for 0°≤𝜃≤360° Solve 𝟓𝒄𝒔𝒄𝜽 𝟑 = 𝟗 𝟒 for 0≤𝜃≤2𝜋 65.2 ˚, 294.8 ˚ 4.1 ˚, 175.9 ˚ 0.83 , 2.31

8.2 Sine and Cosine Curves Objective: To find equations of different sine and cosine curves and to apply these equations. Today: Period & Amplitude

Trig Curves Transformed Graph Sine & Cosine Functions

Variations of the Basic Graphs We are interested in the graphs of functions in the form y = A sin B (x – h) + k and y = A cos B (x – h) + k where A, B, h, and k are all constants. These constants have the effect of translating, reflecting, stretching, and shrinking the basic graphs.

Amplitude = 𝑎 a ≠0 b = 2𝜋 𝑝𝑒𝑟𝑖𝑜𝑑 Period = 2𝜋 𝑏 y = a sin b(x – h) + k and y = a cos b(x – h) + k Amplitude = 𝑎 a ≠0 b = 2𝜋 𝑝𝑒𝑟𝑖𝑜𝑑 Period = 2𝜋 𝑏 Copyright © 2009 Pearson Education, Inc.

Equation of a Sine Function Amplitude Period Complete Cycle

Draw one cycle of the function’s graph. Amplitude Period

Draw one cycle of the function’s graph. Amplitude Period

Draw one cycle of the function’s graph. Amplitude Period

Equation of a Cosine Function Amplitude Period Complete Cycle

Draw one cycle of the function’s graph. Amplitude Period

Draw one cycle of the function’s graph. Amplitude Period

Draw one cycle of the function’s graph. Amplitude Period

Reflection over x-axis Graph the function. Amplitude Period Reflection over x-axis

Reflection over x-axis Graph the function. Amplitude Period Reflection over x-axis

Give the period and amplitude of each function: Graph each equation: 𝑦=4𝑐𝑜𝑠2𝑥 2. 𝑦=3𝑠𝑖𝑛 1 2 𝑥 3. 𝑦=5𝑠𝑖𝑛 2𝜋 7 𝑥 4. 𝑦=6𝑐𝑜𝑠 2𝜋 3 𝑡 Graph each equation: 𝑦=3𝑠𝑖𝑛 𝜋 2 𝑥 𝑦=−2𝑐𝑜𝑠 1 2 𝑥 𝝅,𝟒 𝟒𝝅,𝟑 𝟕,𝟓 𝟑,𝟔

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