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Warm Up Identify the Vertical Shift, Amplitude, Maximum and Minimum, Period, and Phase Shift of each function Y = 3 + 6cos(8x) Y = 2 - 4sin( x + 3) Y = 3cos(x ) -2
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Addition and subtraction of sine, cosine and tangent
We will be able to use the addition and subtraction formulas of sine, cosine, and tangent to determine exact trigonometric values
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EXAMPLES Example 1 Calculate sin(75°) by applying the addition formula: sin(75°) = sin(30°)*cos(45°) + cos(30°)*sin(45°) =
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Calculate cos(75°) by applying the addition formula:
EXAMPLE 2 Calculate cos(75°) by applying the addition formula: Note that 75° = 30° + 45°. cos(75°) = cos(30°)*cos(45°) - sin(30°)*sin(45°) = ???
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Answer to Example 2
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Explore What is the solution to sin(30) + sin(60)?
Is it the same as the solution of sin(90)? What about cos(55) – cos(10) And cos(45)
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+/- of Sine In order to solve addition and subtraction of sine function, we have to set them up correctly Sin(x + y) = sin(x)cos(y) + cos(x)sin(y) Sin(x - y) = sin(x)cos(y) - cos(x)sin(y)
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Addition Example What is the solution of sin(90)?
Lets try and break it down into 30° and 60°
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Subtraction Example Lets evaluate sin(15°)
Now lets break it down into 45° and 30° and check:
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Practice on your own What is sin(135°)?
What if we break it down into 100° and 35°? What if we make it 140° and 5°
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+/- of Cosine Cosine follows similar but different rules
cos(x + y) = cos(x)cos(y) - sin(x)sin(y) cos(x - y) = cos(x)cos(y) + sin(x)sin(y)
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Addition example Cos(60°) = What if we break it down into 30° and 30°?
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Subtraction example What is cos(90°) Break it down into 100° and 10°
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Your turn Cos of 120°? Break it down into 100° and 20°
Now try it as 150° and 30°
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Last one……TAN
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Example Tan(50) = What if we break it down into 25° and 25°
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Your turn Tan(300) = What if we break it down into 360° and 60°?
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Cumulative practice Find the sin(210°) using 100 ° and 110°
Find the cos(210°) using 350° and 140° Find the tan(210°) using 150° and 60°
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