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1.4 Reference Angles

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Definition: Reference Angle - A positive acute angle found between the terminal side of an angle in standard position and the x-axis.

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Examples: Determine the reference angle of the following angles: = 65 Let’s draw a picture: Let’s calculate the reference angle… Terminal side Reference Angle ’ = 65 Let’s see where the reference angle is…

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Examples: Determine the reference angle of the following angles: = 192 Let’s draw a picture: Let’s calculate the reference angle… Reference Angle ’ = 192- 180 ’ = 12 Terminal side Let’s see where the reference angle is…

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Examples: Determine the reference angle of the following angles: = 119 Let’s draw a picture: Let’s calculate the reference angle… Terminal side ’ = 180- 119 Reference Angle ’ = 61 Let’s see where the reference angle is…

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Examples: Determine the reference angle of the following angles: = 341 Let’s draw a picture: Let’s calculate the reference angle… Reference Angle ’ = 360- 341 Terminal side ’ = 19 Let’s see where the reference angle is…

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Examples: Determine the reference angle of the following angles: = /9 Let’s draw a picture: Let’s calculate the reference angle… Reference Angle Terminal side ’ = /9 Let’s see where the reference angle is…

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Examples: Determine the reference angle of the following angles: = 7/5 Let’s draw a picture: Let’s calculate the reference angle… Reference Angle ’ = 7/5 - ’ = 2/5 Terminal side Let’s see where the reference angle is…

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Examples: Determine the reference angle of the following angles: We need to first find a coterminal angle: = -3/8 Let’s draw a picture: -3/8 + 2 -3/8 + 16/8 13/8 Let’s calculate the reference angle… Reference Angle Terminal side Let’s see where the reference angle is… ’ = 2 - 13/8 ’ = 3/8

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**1.4 Working with Reference Angles**

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**Now that you have been exposed to the concept of reference angle, it is time to finish this section.**

We can now determine trigonometric values of non-acute angles.

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Example: Let (-3,4) be a point on the terminal side of (in standard position). Determine the sine, cosine and tangent of . Step 1: Draw a picture of the situation. Step2: Draw the reference triangle (drop the altitude to the x-axis). Then determine the lengths of the sides of the reference triangle. Step 3: Use the non-unit circle definitions to determine the values of the trig. functions requested. Determine the radius of the circle passing through the point referenced. (-3,4) 5 4 3 sin = cos = tan = 4 5 -3 5 -4 3

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Example: Let (-4,-6) be a point on the terminal side of (in standard position). Determine the sine, cosine, tangent, cosecant, secant and cotangent of . Step 1: Draw a picture of the situation. Step2: Draw the reference triangle (drop the altitude to the x-axis). Then determine the lengths of the sides of the reference triangle. Step 3: Use the non-unit circle definitions to determine the values of the trig. functions requested. Determine the radius of the circle passing through the point referenced. 4 sin = cos = tan = csc = sec = cot = -3√13 13 -2√13 13 3 2 6 (-4,-6) 2√13 -√13 3 2 3 -√13 2

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All Star Trig. Class (−,+) (+,+) (−,−) (+,−)

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