splash screen. lesson menu five-minute check (over lesson 4-2) then/now new vocabulary key concept:...
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Five-Minute Check (over Lesson 4-2)
Then/Now
New Vocabulary
Key Concept: Trigonometric Functions of Any Angle
Example 1: Evaluate Trigonometric Functions Given a Point
Key Concept: Common Quadrantal Angles
Example 2: Evaluate Trigonometric Functions of Quadrantal Angles
Key Concept: Reference Angle Rules
Example 3: Find Reference Angles
Key Concept: Evaluating Trigonometric Functions of Any Angle
Example 4: Use Reference Angles to Find Trigonometric Values
Example 5: Use One Trigonometric Value to Find Others
Example 6: Real-World Example: Find Coordinates Given a Radius and an Angle
Key Concept: Trigonometric Functions on the Unit Circle
Example 7: Find Trigonometric Values Using the Unit Circle
Key Concept: Periodic Functions
Example 8: Use the Periodic Nature of Circular Functions
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Over Lesson 4-2
Write 62.937˚ in DMS form.
A. 62°54'13"
B. 63°22'2"
C. 62°54'2"
D. 62°56'13.2"
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Over Lesson 4-2
Write 96°42'16'' in decimal degree form to the nearest thousandth.
A. 96.704o
B. 96.422o
C. 96.348o
D. 96.259o
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Over Lesson 4-2
Write 135º in radians as a multiple of π.
A.
B.
C.
D.
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Over Lesson 4-2
A. 240o
B. –60o
C. –120o
D. –240o
Write in degrees.
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Over Lesson 4-2
Find the length of the intercepted arc witha central angle of 60° in a circle with a radius of15 centimeters. Round to the nearest tenth.
A. 7.9 cm
B. 14.3 cm
C. 15.7 cm
D. 19.5 cm
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You found values of trigonometric functions for acute angles using ratios in right triangles. (Lesson 4-1)
• Find values of trigonometric functions for any angle.
• Find values of trigonometric functions using the unit circle.
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• quadrantal angle
• reference angle
• unit circle
• circular function
• periodic function
• period
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Evaluate Trigonometric Functions Given a Point
Let (–4, 3) be a point on the terminal side of an angle θ in standard position. Find the exact values of the six trigonometric functions of θ.
Pythagorean Theorem
x = –4 and y = 3
Use x = –4, y = 3, and r = 5 to write the six trigonometric ratios.
Take the positive square root.
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Evaluate Trigonometric Functions Given a Point
Answer:
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Let (–3, 6) be a point on the terminal side of an angle Ө in standard position. Find the exact values of the six trigonometric functions of Ө.
A.
B.
C.
D.
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Evaluate Trigonometric Functions of Quadrantal Angles
A. Find the exact value of cos π. If not defined, write undefined.
The terminal side of π in standard position lies on the negative x-axis. Choose a point P on the terminal side of the angle. A convenient point is (–1, 0) because r = 1.
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Evaluate Trigonometric Functions of Quadrantal Angles
Answer: –1
x = –1 and r = 1
Cosine function
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Evaluate Trigonometric Functions of Quadrantal Angles
B. Find the exact value of tan 450°. If not defined, write undefined.
The terminal side of 450° in standard position lies on the positive y-axis. Choose a point P(0, 1) on the terminal side of the angle because r = 1.
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Evaluate Trigonometric Functions of Quadrantal Angles
Answer: undefined
y = 1 and x = 0
Tangent function
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Evaluate Trigonometric Functions of Quadrantal Angles
C. Find the exact value of . If not defined, write undefined.
The terminal side of in standard position lies
on the negative y-axis. The point (0, –1) is convenient
because r = 1.
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Evaluate Trigonometric Functions of Quadrantal Angles
Answer: 0
x = 0 and y = –1
Cotangent function
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A. –1
B. 0
C. 1
D. undefined
Find the exact value of sec If not defined, write undefined.
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Find Reference Angles
A. Sketch –150°. Then find its reference angle.
A coterminal angle is –150° + 360° or 210°. The terminal side of 210° lies in Quadrant III. Therefore, its reference angle is 210° – 180° or 30°.
Answer: 30°
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Find Reference Angles
Answer:
The terminal side of lies in Quadrant II. Therefore,
its reference angle is .
B. Sketch . Then find its reference angle.
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Find the reference angle for a 520o angle.
A. 20°
B. 70°
C. 160°
D. 200°
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Use Reference Angles to Find Trigonometric Values
A. Find the exact value of .
Because the terminal side of lies in Quadrant III, the
reference angle
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Use Reference Angles to Find Trigonometric Values
Answer:
In Quadrant III, sin θ is negative.
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Use Reference Angles to Find Trigonometric Values
B. Find the exact value of tan 150º.
Because the terminal side of θ lies in Quadrant II, the reference angle θ' is 180o – 150o or 30o.
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Use Reference Angles to Find Trigonometric Values
Answer:
tan 150° = –tan 30° In Quadrant II, tan θ is negative.
tan 30°
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Use Reference Angles to Find Trigonometric Values
C. Find the exact value of .
A coterminal angle of which lies in
Quadrant IV. So, the reference angle
Because cosine and secant are reciprocal functions
and cos θ is positive in Quadrant IV, it follows that
sec θ is also positive in Quadrant IV.
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Use Reference Angles to Find Trigonometric Values
In Quadrant IV, sec θ is positive.
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Use Reference Angles to Find Trigonometric Values
Answer:
CHECK You can check your answer by using a graphing calculator.
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A.
B.
C.
D.
Find the exact value of cos .
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Use One Trigonometric Value to Find Others
To find the other function values, you must find the coordinates of a point on the terminal side of θ. You know that sec θ is positive and sin θ is positive, so θ must lie in Quadrant I. This means that both x and y are positive.
Let , where sin θ > 0. Find the exact
values of the remaining five trigonometric
functions of θ.
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Use One Trigonometric Value to Find Others
Because sec =
and x = 5 to find y.
Take the positive square root.
Pythagorean Theorem
r = and x = 5
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Use One Trigonometric Value to Find Others
Use x = 5, y = 2, and r = to write the other five trigonometric ratios.
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Use One Trigonometric Value to Find Others
Answer:
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Let csc θ = –3, tan θ < 0. Find the exact values o the five remaining trigonometric functions of θ.
A.
B.
C.
D.
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ROBOTICS A student programmed a 10-inch long robotic arm to pick up an object at point C and rotate through an angle of 150° in order to release it into a container at point D. Find the position of the object at point D, relative to the pivot point O.
Find Coordinates Given a Radius and an Angle
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Find Coordinates Given a Radius and an Angle
Cosine ratio
= 150° and r = 10
cos 150° = –cos 30°
Solve for x.
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Find Coordinates Given a Radius and an Angle
Sin ratio
θ = 150° and r = 10
sin 150° = sin 30°
Solve for y.5 = y
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Find Coordinates Given a Radius and an Angle
Answer: The exact coordinates of D are .
The object is about 8.66 inches to the left of
the pivot point and 5 inches above the pivot
point.
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CLOCK TOWER A 4-foot long minute hand on a clock on a bell tower shows a time of 15 minutes past the hour. What is the new position of the end of the minute hand relative to the pivot point at 5 minutes before the next hour?
A. 6 feet left and 3.5 feet above the pivot point
B. 3.4 feet left and 2 feet above the pivot point
C. 3.4 feet left and 6 feet above the pivot point
D. 2 feet left and 3.5 feet above the pivot point
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Find Trigonometric Values Using the Unit Circle
Definition of sin tsin t = y
Answer:
A. Find the exact value of . If undefined,
write undefined.
corresponds to the point (x, y) = on
the unit circle.
y = . sin
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Find Trigonometric Values Using the Unit Circle
Answer:
cos t = x Definition of cos t
cos
corresponds to the point (x, y) = on the
unit circle.
B. Find the exact value of . If undefined,
write undefined.
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Find Trigonometric Values Using the Unit Circle
Definition of tan t.
C. Find the exact value of . If undefined,
write undefined.
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Find Trigonometric Values Using the Unit Circle
Simplify.
Answer:
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Find Trigonometric Values Using the Unit Circle
D. Find the exact value of sec 270°. If undefined,
write undefined.
270° corresponds to the point (x, y) = (0, –1) on the unit circle.
Therefore, sec 270° is undefined.
Answer: undefined
Definition of sec t
x = 0 when t = 270°
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A.
B.
C.
D.
Find the exact value of tan . If undefined, write undefined.
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Use the Periodic Nature of Circular Functions
cos t = x and x =
A. Find the exact value of .
Rewrite as the sum of a
number and 2π.
+ 2π map to the same
point (x, y) = on the
unit circle.
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Use the Periodic Nature of Circular Functions
Answer:
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B. Find the exact value of sin(–300).
sin (–300o) = sin (60o + 360o(–1)) Rewrite –300o as the sum of a number and an integer multiple of 360o.
Use the Periodic Nature of Circular Functions
= sin 60o 60o and 60o
+ 360o(–1) map to the
same point
(x, y) =
on the unit
circle.
![Page 56: Splash Screen. Lesson Menu Five-Minute Check (over Lesson 4-2) Then/Now New Vocabulary Key Concept: Trigonometric Functions of Any Angle Example 1: Evaluate](https://reader037.vdocument.in/reader037/viewer/2022102906/56649d1b5503460f949f0bf0/html5/thumbnails/56.jpg)
Use the Periodic Nature of Circular Functions
= sin t = y
and y = when t =
60o.
Answer:
![Page 57: Splash Screen. Lesson Menu Five-Minute Check (over Lesson 4-2) Then/Now New Vocabulary Key Concept: Trigonometric Functions of Any Angle Example 1: Evaluate](https://reader037.vdocument.in/reader037/viewer/2022102906/56649d1b5503460f949f0bf0/html5/thumbnails/57.jpg)
Use the Periodic Nature of Circular Functions
C. Find the exact value of .
Rewrite as the sum of a
number and 2 and an integer
multiple of π.
map to the
same point (x, y) =
on the unit circle.
![Page 58: Splash Screen. Lesson Menu Five-Minute Check (over Lesson 4-2) Then/Now New Vocabulary Key Concept: Trigonometric Functions of Any Angle Example 1: Evaluate](https://reader037.vdocument.in/reader037/viewer/2022102906/56649d1b5503460f949f0bf0/html5/thumbnails/58.jpg)
Use the Periodic Nature of Circular Functions
Answer:
![Page 59: Splash Screen. Lesson Menu Five-Minute Check (over Lesson 4-2) Then/Now New Vocabulary Key Concept: Trigonometric Functions of Any Angle Example 1: Evaluate](https://reader037.vdocument.in/reader037/viewer/2022102906/56649d1b5503460f949f0bf0/html5/thumbnails/59.jpg)
A. 1
B. –1
C.
D.
Find the exact value of cos
![Page 60: Splash Screen. Lesson Menu Five-Minute Check (over Lesson 4-2) Then/Now New Vocabulary Key Concept: Trigonometric Functions of Any Angle Example 1: Evaluate](https://reader037.vdocument.in/reader037/viewer/2022102906/56649d1b5503460f949f0bf0/html5/thumbnails/60.jpg)