Trigonometric Functions
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Transcript of Trigonometric Functions
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Trigonometric FunctionsOf Any Angle
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Unit Circle Rationale
Recall that when using the unit circle to evaluate the value of a trig function, cos θ = x and sin θ = y. What we didn’t point out is that since the radius (hypotenuse) is 1, the trig
values are really cos θ = and sin θ = .
S So what if the radius (hypotenuse) is not 1?
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Trig Function of Any Angle
Let θ be an angle whose terminal side contains the point (4, 3). Find sin θ, cos θ, and tan θ.
We must first find the hypotenuse (r) by using the Pythagorean Theorem.
sin 𝜃=𝑜𝑝𝑝h𝑦𝑝 =
35 cos 𝜃=
𝑎𝑑𝑗h𝑦𝑝=
45
tan𝜃=𝑜𝑝𝑝𝑎𝑑𝑗 =
34
𝒓=√𝒙𝟐+𝒚𝟐
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Trig Function of Any Angle
The six trig functions of a unit circle, where , are:
sin 𝜃=𝑜𝑝𝑝h𝑦𝑝 =
𝑦1 =
𝒚𝒓 csc 𝜃=
h𝑦𝑝𝑜𝑝𝑝=
1𝑦=
𝒓𝒚
cos𝜃=𝑎𝑑𝑗h𝑦𝑝=
𝑥1 =
𝒙𝒓 sec𝜃=
h𝑦𝑝𝑎𝑑𝑗 =
1𝑥=
𝒓𝒙
tan𝜃=𝑜𝑝𝑝𝑎𝑑𝑗 =
𝒚𝒙 cot 𝜃=
𝑎𝑑𝑗𝑜𝑝𝑝=
𝒙𝒚
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Trig Function of Any Angle
The six trig functions of any angle, where
sin 𝜃=𝑜𝑝𝑝h𝑦𝑝 =
𝒚𝒓
csc 𝜃=h𝑦𝑝𝑜𝑝𝑝=
𝒓𝒚
cos𝜃=𝑎𝑑𝑗h𝑦𝑝=
𝒙𝒓 sec𝜃=
h𝑦𝑝𝑎𝑑𝑗 =
𝒓𝒙
tan𝜃=𝑜𝑝𝑝𝑎𝑑𝑗 =
𝒚𝒙 cot 𝜃=
𝑎𝑑𝑗𝑜𝑝𝑝=
𝒙𝒚
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Quadrant IQuadrant II
Quadrant III Quadrant IV
ALLSTUDENTS
TAKE CALCULUS
PositiveALL
Positivesin, csc
Positivetan, cot
Positivecos, sec
¿¿
(− ,−) (+ ,−)
90 °
180 °
270 °
0 °360 °
2𝜋
The Signs of the Trig Functions
Always look for the quadrant where the terminal side of the angle is located.
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P
Let be a point on the terminal side of . Find each of the six trigonometric functions of . Note: We must first find by using the Pythagorean Theorem.
𝐬𝐢𝐧 𝜽=𝒚𝒓 𝐜𝐬𝐜𝜽=
𝒓𝒚
𝐜𝐨𝐬𝜽=𝒙𝒓 𝐬𝐞𝐜 𝜽=
𝒓𝒙
𝐭𝐚𝐧 𝜽=𝒚𝒙 𝐜𝐨𝐭 𝜽=
𝒙𝒚
(−𝟑 ,𝟒)
−𝟑
𝟒 𝟓
𝒓=√𝒙𝟐+𝒚𝟐𝒓=√ (−𝟑 )𝟐+(𝟒 )𝟐 𝒓=√𝟗+𝟏𝟔=√𝟐𝟓=¿
¿𝟒𝟓 ¿
𝟓𝟒
¿−𝟑𝟓
Notice that is an obtuse angle. In which quadrant does the terminal side lie?
¿−𝟓𝟑
¿−𝟒𝟑 ¿−𝟑
𝟒𝑰𝑰
Check for the validity of the signs in the answers.
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Step-by-step, inch-by-inch What is the coordinate? Identify x and y.
Find r by using the equation
Substitute the values for x, y, and r into the six trig identities of any triangle.
Check the signs of your answers by using, “ALL STUDENTS TAKE CALCULUS”. (The terminal side of the angle determines which quadrant to use.)
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Let be a point on the terminal side of . Find each of the six trigonometric functions of . Note: We must first find by using the Pythagorean Theorem.
𝐬𝐢𝐧 𝜽=𝒚𝒓 𝐜𝐬𝐜𝜽=
𝒓𝒚
𝐜𝐨𝐬𝜽=𝒙𝒓 𝐬𝐞𝐜 𝜽=
𝒓𝒙
𝐭𝐚𝐧 𝜽=𝒚𝒙 𝐜𝐨𝐭 𝜽=
𝒙𝒚
𝒓=√𝒙𝟐+𝒚𝟐𝒓=√ (−𝟑 )𝟐+(−𝟒 )𝟐𝒓=√𝟗+𝟏𝟔=√𝟐𝟓=¿
¿−𝟒𝟓 ¿−𝟓
𝟒
¿−𝟑𝟓 ¿−𝟓
𝟑
¿𝟒𝟑 ¿
𝟑𝟒
(−𝟑 ,−𝟒)
𝟓
−𝟑
−𝟒
In which quadrant does the terminal side lie?
𝑰𝑰𝑰
Check for the validity of the signs in the answers.
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Let be a point on the terminal side of . Find each of the six trigonometric functions of . Note: We must first find by using the Pythagorean Theorem.
𝐬𝐢𝐧 𝜽=𝒚𝒓 𝐜𝐬𝐜𝜽=
𝒓𝒚
𝐜𝐨𝐬𝜽=𝒙𝒓 𝐬𝐞𝐜 𝜽=
𝒓𝒙
𝐭𝐚𝐧 𝜽=𝒚𝒙 𝐜𝐨𝐭 𝜽=
𝒙𝒚
𝒓=√𝒙𝟐+𝒚𝟐𝒓=√ (𝟑 )𝟐+(−𝟒 )𝟐𝒓=√𝟗+𝟏𝟔=√𝟐𝟓=¿
¿−𝟒𝟓 ¿−𝟓
𝟒
¿𝟑𝟓 ¿
𝟓𝟑
¿−𝟒𝟑 ¿−𝟑
𝟒
In which quadrant does the terminal side lie?
(𝟑 ,−𝟒)
In which quadrant does the terminal side lie?
𝟑
−𝟒𝟓
𝐼𝑉
Check for the validity of the signs in the answers.
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Trig Functions of Quadrantal AnglesEvaluate the sine function at the following four quadrantal angles.
𝒓=√𝒙𝟐+𝒚𝟐
𝒓=√𝟏𝟐+𝟎𝟐=𝟏sin 𝜃=
𝑦𝑟
First, we must find r.
sin 0 °=01=𝟎 sin 𝜋
2 =11=𝟏
sin 𝜋=01 =𝟎 sin 3𝜋
2 =− 11=−𝟏
Step-by-StepInch-by-Inch
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Trig Functions of Quadrantal AnglesEvaluate the tangent function at the following four quadrantal angles.
tan𝜃=𝑦𝑥
Since we only need x and y to find tangent, we don’t need to find r.
tan 0 °=01 =𝟎tan 𝜋2 =
10=𝒖𝒏𝒅𝒆𝒇𝒊𝒏𝒆𝒅
tan 𝜋=0−1=𝟎 tan 3𝜋
2 =−10 =𝒖𝒏𝒅𝒆𝒇𝒊𝒏𝒆𝒅
Would the answers change if the angles were expressed as degrees? No
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Quadrant IQuadrant II
Quadrant III Quadrant IV
ALLSTUDENTS
TAKE CALCULUS
PositiveALL
Positivesin, csc
Positivetan, cot
Positivecos, sec
¿¿
(− ,−) (+ ,−)
90 °
180 °
270 °
0 °360 °
2𝜋
The Signs of the Trig FunctionsIf and , name the quadrant in which angle lies.
In which quadrants is tangent negative?
𝐼𝐼 𝑎𝑛𝑑 𝐼𝑉
In which quadrants is cosine positive?
𝐼 𝑎𝑛𝑑 𝐼𝑉Angle lies in quadrant IV
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Quadrant IQuadrant II
Quadrant III Quadrant IV
ALLSTUDENTS
TAKE CALCULUS
PositiveALL
Positivesin, csc
Positivetan, cot
Positivecos, sec
¿¿
(− ,−) (+ ,−)
90 °
180 °
270 °
0 °360 °
2𝜋
The Signs of the Trig FunctionsIf and , name the quadrant in which angle lies.
In which quadrants is sine negative?
In which quadrants is cosine negative?
𝐼𝐼𝐼 𝑎𝑛𝑑 𝐼𝑉
𝐼𝐼 𝑎𝑛𝑑 𝐼𝐼𝐼
Angle lies in quadrant III
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Evaluating Trigonometric FunctionsGiven and , find and .
Step 1: Find the quadrant where the angle lies. Tangent is negative in quadrants II and IV. Cosine is positive in quadrants I and IV. The angle lies in quadrant IV, where the coordinate is
Step 2: Find x and y.
Step 3: Find r.tan𝜃=
𝑦𝑥 𝑥=3∧𝑦=−2
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Evaluating Trigonometric Functions(Cont’d.) Given and , find and .
Step 3: We now know that and .
Check: cosine is positive in quadrant IV.
Check: Cosecant is negative in quadrant IV.
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Evaluating Trigonometric FunctionsGiven and , find and .
Step 1: Find the quadrant where the angle lies. Tangent is negative in quadrants II and IV. Cosine is negative in quadrants II and III. The quadrant is II, therefore, the x and y-coordinate for tangent is .
Step 2: Find x and y. 𝑥=−3 𝑎𝑛𝑑 𝑦=1
Step 3: Find r.
𝑟=√𝑥2 +𝑦 2=√32+ (−1 )2=√9+1=√10
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Evaluating Trigonometric Functions(Cont’d.) Given and , find and .
Step 3: We now know that and .
sin 𝜃= 𝑦𝑟 = 1
√1 0= 1
√10∙ √1 0√1 0
=1√101 0
=√𝟏𝟎𝟏𝟎
sec𝜃= 𝑟𝑥=√10
−3=− √𝟏𝟎
𝟑
Check: Sine is positive in quadrant II.
Check: Cosecant is negative in quadrant II.
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Reference Angles
The reference angle is the positive acute angle formed by the terminal side of the angle and the x-axis.
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Reference Angles
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Reference Angles
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Find the reference angle , for
𝜽′=𝟏𝟓°
𝜽′=𝝅𝟔
𝜽′=𝟒𝟓°
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Finding Reference Angles for Angles Greater Than 360° or Less Than
Find a positive angle less than 360° or that is coterminal with the given angle.
Draw angle in standard position.
Use the drawing to find the reference angle for the given angle. The positive acute angle formed by the terminal side of and the x-axis is the reference angle.
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Find the reference angle , for
𝜽′=𝟒𝟎°
𝜽′=𝝅𝟑
𝜽′=𝝅𝟔
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Evaluating Trig Functions Using Reference Angles
Use reference angles to find the exact value of .
Step 1: Find the reference angle of , which is
Step 2: Find the quadrant that lies in, which is quadrant II.
Step 3: The sine function is positive in quadrant II.
Therefore,
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Evaluating Trig Functions Using Reference Angles
Use reference angles to find the exact value of .
Step 1: Find the reference angle of , which is
Step 2: Find the quadrant that lies in, which is quadrant III.
Step 3: The cosine function is negative in quadrant III.
Therefore,
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Evaluating Trig Functions Using Reference Angles
Use reference angles to find the exact value of .
Step 1: Find the reference angle of , which is
Step 2: Find the quadrant that lies in, which is quadrant IV.
Step 3: The cotangent function is negative in quadrant IV.
Therefore,
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