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Chapter 4.3 Trigonometric ratiosPages 191 2054
Recall:Ratios for the three primary trig ratios:
SOH CAH TOA (Some Old Hippie Came A Hoppin' Through Our Apartment)
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Recall from 521B, we can use SOH CAH TOA and the following diagram to develop ratios to determine any trig ratio for any circle:
y
x
r
reciprocal of the: sine ratio is the cosecant ratio >
cosine ratio is the secant ratio >
tangent ratio is the cotangent ratio >
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If (x, y) is part of the unit circle, the following is true:
y
x
r1 Therefore, if a point is on the unit circle
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Example 1Page 193
Determine the Trigonometric Ratios for Angles in the Unit Circle
the terminal arm of an angle θ in standard position.a) Draw a diagram to model the situation.b) Determine the values of the six trigonometric ratios for θ. Express answers in lowest terms.
The point lies at the intersection of the unit circle and
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The point lies at the intersection of the unit circle
and the terminal arm of an angle θ in standard position.a) Draw a diagram to model the situation.b) Determine the values of the six trigonometric ratios for θ. Express your answers in lowest terms.
a)
Answer
Example 1: Your TurnPage 194
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As a result, the unit circle gives rise to the following:
θ 0
sinθ
cosθ
tanθ
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Using the values from the first quadrant, the CAST rule, and reference angles, we can determine the following:
Example 2Page 194
Exact Values for Trigonometric RatiosDetermine the exact value for each. Draw diagrams to illustrate your answers.
c) sec 315° d) cot 270°
a) b)
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Example 2: Your TurnPage 195
Draw diagrams to help you determine the exact value of each trigonometric ratio.a) b)
c) sin (–300°) d) sec 60°
Answer
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Approximating values for the trig ratios:
Example 3Page 196
Approximate Values for Trigonometric RatiosDetermine the approximate value for each trigonometric ratio. Give youranswers to four decimal places.a) b) cos 260°
c) sin 4.2 d) csc (–70°)
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Example 3: Your TurnPage 197
What is the approximate value for each trigonometric ratio? Roundanswers to four decimal places. Justify the sign of each answer.a) sin 1.92b) tan (–500°)c) sec 85.4°d) cot 3
Answer
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Example 5Page 200
Calculating Trigonometric Values for Points Not on the Unit CircleThe point A(–4, 3) lies on the terminal arm of an angle θ in standardposition. What is the exact value of each trigonometric ratio for θ?
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Example 5: Your TurnThe point D(–5, –12) lies on the terminal arm of an angle θ in standardposition. What is the exact value of each trigonometric ratio for θ?
Answer
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Additional Example: Determine the exact values for the remaining five trig ratios under the given conditions.
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A mnemonic device, such as 'All Students Take Chemistry' can help one remember the sign of the trig ratio when an angle terminates in specific quadrants.
AS
T C
Quadrant I All functions are positive.Quadrant II Sine and its reciprocal are positive.Quadrant III Tangent and its reciprocal are positive.Quadrant IV Cosine and its reciprocal are positive.
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More Examples:1. Determine the quadrant(s) in which may be terminate, if it is an angle in standard position if:
a)
b) and
c) and
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It may be helpful to notice that a trig ratio of a large angle may equal the same trig ratio of its reference angle or be equal to the negative value of the same trig ratio of its reference angle.
2. Express the given quantity using the same trig ratio and its reference angle.
a)
b)
c)
3. Find the exact value of
