Trigonometric Functions

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Trigonometric Functions Of Any Angle

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Trigonometric Functions. Of Any Angle. 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 - PowerPoint PPT Presentation

Transcript of Trigonometric Functions

Page 1: Trigonometric Functions

Trigonometric FunctionsOf Any Angle

Page 2: Trigonometric Functions

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?

Page 3: Trigonometric Functions

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

Page 4: Trigonometric Functions

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

Page 5: Trigonometric Functions

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

Page 6: Trigonometric Functions

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.

Page 7: Trigonometric Functions

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.

Page 8: Trigonometric Functions

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.)

Page 9: Trigonometric Functions

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.

Page 10: Trigonometric Functions

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

Page 12: Trigonometric Functions

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

Page 13: Trigonometric Functions

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

Page 14: Trigonometric Functions

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

Page 15: Trigonometric Functions

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

Page 16: Trigonometric Functions

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.

Page 17: Trigonometric Functions

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

Page 18: Trigonometric Functions

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.

Page 19: Trigonometric Functions

Reference Angles

The reference angle is the positive acute angle formed by the terminal side of the angle and the x-axis.

Page 20: Trigonometric Functions

Reference Angles

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Reference Angles

Page 22: Trigonometric Functions

Find the reference angle , for

𝜽′=𝟏𝟓°

𝜽′=𝝅𝟔

𝜽′=𝟒𝟓°

Page 23: Trigonometric Functions

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.

Page 24: Trigonometric Functions

Find the reference angle , for

𝜽′=𝟒𝟎°

𝜽′=𝝅𝟑

𝜽′=𝝅𝟔

Page 25: Trigonometric Functions

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,

Page 26: Trigonometric Functions

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,

Page 27: Trigonometric Functions

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,

Page 28: Trigonometric Functions