2.1 Angle Measures Name: ________________________________

Obj.: I will be able to identify angle types, convert between degrees and radians for angle measures, identify coterminal angles, find the length of an intercepted arc, and find linear and angular speeds.

VocabularyAngle Initial side Terminal side Vertex Standard positionPositive angles Negative angles Quadrantal angle Degrees Acute angleRight angle Obtuse angle Straight angle Radian Central AngleRadian Measure Coterminal angles Linear Speed Angular Speed

Notes Recognize and use the vocabulary of angles

o An angle (α,β,θ, etc.) is formed by two rays that have a common endpoint. One ray is called the initial side and the other the terminal side.

o Standard position: vertex is at the origin of the coordinate system and the initial ray is on the x-axis.

Sign Positive angles are generated by rotating counterclockwise. Negative angles are generated by rotating clockwise.

Types If the terminal ray is between axes, it lies in that quadrant. If the terminal ray is ON the x-axis or y-axis, it is a quadrantal angle

Use degree measureo If the terminal ray has gone in a complete circle and is back on the initial angle, it has

gone 360°.

o 1 °= 1360

complete rotation

Using radian measureo One radian is the measure of the central angle of a circle that intercepts an arc equal in

length to the radius of the circle.o The radian measure of any central angle is the length of the intercepted arc divided by

the circle’s radius.

θ= lengthof the interceptedarcradius

= sr radians

Convert between degrees and radians

o To convert degrees to radians, multiply degrees by π radians180°

2.1 Angle Measures Name: ________________________________

o To convert radians to degrees, multiply radians by 180°

π radians

Draw angles in standard positiono It is important to learn to recognize angles in the coordinate plane by radians – start

learning the unit circle!

Find coterminal angleso Two angles with the same initial and terminal sides but possibly different rotations are

called coterminal angles. o Every angle has infinitely many coterminal angles.

Increasing or decreasing the degree measure by an integer multiple of 360° (or a radian measure by a multiple of 2π) results in a coterminal angle

Positive and negative angles whose absolute values add up to 2π are coterminal angles.

Find the length of a circular arc

o Can use the radian measure formula θ= sr

to find the length of the intercepted arc, s.

s=rθ Use linear and angular speed to describe circular motion

o If a point is in motion on a circle of radius r through an angle of θ radians in time t, then

its linear speed is v= st

o Its angular speed is ω=θt

o Linear speed is the product of the radius and the angular speed.st=rθt

=r θt

v=r ω

2.1 Angle Measures Name: ________________________________

Practice

Find the radian measure of the central angle of a circle with the given radius and arc length.1. r = 5, s = 11 2. r = 9, s = 15

Convert the following angles from degrees to radians.3. 135° 4. 117°

Convert the following angles from radians to degrees.5. 6π

56. 10π

9

Draw the given angle in standard position and state the quadrant in which the angle lies. Do not convert to degrees.7. 5π

38. −π

3

Find a positive angle less than 360° that is coterminal with the given angle.9. −80 ° 10. −340 °

Find a positive angle less than 2π that is coterminal with the given angle.11.

29π8

12. 16π5

Find the length of the arc, s, on a circle of radius r intercepted by a central angle θ. Express arc length in terms of π. Then round your answer to two decimal places.13.

r = 5, θ = 325° 14. r = 10, θ = 200°

15. A wind machine used to generate electricity has blades that are 10 feet in length. The propeller is rotating at four revolutions per second. Find the linear speed, in feet per second, of the tips of the blades.

2.2 Right Triangle Trigonometry Name: ________________________________

Obj.: I will be able to calculate trigonometric functions of acute angles. I will be able to solve for sides and angles of right triangles.

VocabularyTrigonometry Hypotenuse Sine of θ Cosine of θ Tangent of θCosecant of θ Secant of θ Cotangent of θ SOHCAHTOA Pythagorean Theorem

Notes Use right triangles to evaluate trigonometric functions

o _____________________ means “measurement of triangles” and is used in navigation, building, and engineering.

o There are ______ main trigonometric functions and __________ inverse functions. _____________ are measures of _________ angles in ___________ triangles ________________ of _____________ in triangle is very important.

___________ of the functions are ____________ rather than ___________. ________________ represent _________________ of sides in a right triangle.

Primary Name Abbreviation Reciprocal Name Abbreviation Inverse Name Abbreviationsine cosecant arcsine

cosine secant arccosinetangent cotangent arctangent

o _________________ and ____________________ functions are used to find ____________ while____________ functions are used to find ___________ measures.

o The trigonometric function values of ____ depend only on the _________ of angle θ and _______ on the size of the ____________________.

2.2 Right Triangle Trigonometry Name: ________________________________

o Pythagorean Theorem The ________ of the _______________ of the _______ of a right triangle _____

the square of the length of the hypotenuse (____________________). ________________ all __________ sides to evaluate all ________ functions.

o Remember to ________________ the ____________________ when necessary.

Find function values for 30°( π6 ), 45°( π4 ), and 60° ( π3 )o Start studying the outside of the Unit Circle for sine and cosine of common angles.o If you must find the reciprocal, do so ________________ rationalizing the denominator.

θ 30°¿( π6 ), 45°¿( π4 ) 60° ¿( π3 )sin θ

cos θ

tan θ

Practice

Use the Pythagorean theorem to find the missing side, then determine the values of each of the six main trigonometric functions. Give answer in simplified fraction form.

1. c=? 2. sin θ=¿ 3. cosθ=¿

4. tanθ=¿ 5. csc θ=¿ 6. secθ=¿

7. cot θ=¿

For each for the following triangles, find the missing variable.8. 9.

2.2 Right Triangle Trigonometry Name: ________________________________

10. 11.

2.3 Trigonometric Identities Name: ________________________________

Obj.: I will be able to use trigonometric identities to find the values of the six main trigonometric functions for an angle.

VocabularyTrigonometric Identities

Reciprocal identities

Quotient Identities

Pythagorean Identities

Complements

Cofunctions

Notes Recognize and Use Fundamental Identities

o Trigonometric__________________ are __________________ between trigonometric __________________

o ________________ can be used to find ______________ trigonometric functions.o Reciprocal Identities

o Quotient Identities

o Pythagorean Identities

Use equal ________________ of __________________

o _______ ________________ angles are ________________ if their ______ is 90° or π2

.

o Cofunctions: Any pair of trigonometric functions _____ and _____ for which

o Cofunction Identities

If θ is in _________________, replace _______ with ______

2.3 Trigonometric Identities Name: ________________________________

Practiceθ is an acute angle and sin θ and cos θ are given. Use identities to find tan θ, csc θ, sec θ, and

cot θ. Where necessary, rationalize denominators. sin θ=35,cosθ= 4

5 .

1. tan θ 2. csc θ

3. sec θ 4. cot θ

5. The angle θ is an acute angle and sin θ=49 . Use the Pythagorean identity sin2θ+cos2θ=1 to

find cos θ.

6. The angle θ is an acute angle and cosθ=29 . Use the Pythagorean identity sin2θ+cos2θ=1 to

find sin θ.

Use an identity to find the value of the expressions. Do not use a calculator.7. sin 27 ° csc27 °

8.sec268 °−tan268°

Find a cofunction with the same value as the given expression.

9. sin 7 ° 10.

tan52 °

11. cos ( 2π7 ) 12

. sin( 4 π9 )

2.4 Applications of Trigonometry Name: ________________________________

Obj.: I will understand elevation and depression, be able to identify angles of elevation and depression. I will be able to apply these concepts to real-world applications.

VocabularyAngle of elevation Angle of depression

Notes Use right triangle trigonometry to solve applied problems

o Many _____________________ of right triangle trigonometry involve the __________ made with an ___________________ ___________________ line.

The angle ____________ the ______________________ is called the angle of ___________________.

The angle ____________ the ______________________ is called the angle of ___________________.

Note: the _____________ of the angle can be to the __________ or to the __________.

2.4 Applications of Trigonometry Name: ________________________________

Practice

1. To find the distance across a lake, a surveyor took the measurements in the figure shown. Use these measurements to determine how far it is across the lake.

2. A tower that is 100 feet tall casts a shadow 144 feet long. Find the angle of elevation of the sun to the nearest degree.

3. A plane rises from take-off and flies at an angle of 14° with the horizontal runway. When it has gained 900ft, find the distance, to the nearest foot, the plane has flown.

4. A telephone pole, shown below, is 60 feet tall. A guy wire 81 feet long is attached from the ground to the top of the pole. Find the angle between the wire and the pole to the nearest degree.

5. From the top of a 319 -foot lighthouse, a plane is sighted overhead, and a ship is observed directly below the plane. The angle of elevation of the plane is 26° , and the angle of depression of the ship is 43°. Find a. the distance of the ship from the lighthouse and b. the plane's height above the water. Round to the nearest foot.

2.5 Reference Angles Name: ________________________________

Obj.: I will be able to find a reference angle for obtuse angles. I will be able to use the reference angle to evaluate trigonometric functions of obtuse angles.

VocabularyReference angles

Notes Find Reference Angles

o Trigonometric functions of ________________ _____________ angles and all __________ angles, are evaluated by making use of a ____________ __________ angle.

o _______________angle – the positive acute angle ______ formed by the ________________side of ____ and the ____________.

o Finding reference angles for angles greater than 360° (2π) or less than –360° (–2π) Find a positive angle _____ less than 360° or 2π that is ________________ with

the given angle. ____________ α in _______________ position. Use the drawing to find the ________________ angle for the given angle. The

_______________ acute angle formed by the terminal side of α and the x-axis is the reference _____________.

Use reference angles to evaluate trigonometric functionso The _____________ of the trigonometric functions of a given angle, ¿¿, are the

__________ as the ____________ of the trigonometric functions of the ______________ angle, ______, except possibly for the _______

o Procedure for using reference angles to _________________ trigonometric functions: Find the _______________ reference angle,θ ' , and the ________________

_____________ for θ '. Use the _______________ in which θ lies to __________ the appropriate

___________ to the function value in the first step. Use the signs of the trigonometric functions

o If θ is not a _________________ angle, the _________ of a trigonometric function ______________ on the ________________ in which θ lies.

Quadrant I: _____ ______________ are positive Quadrant II: ____________ and _______________ are positive Quadrant III: ______________ and __________________ are positive Quadrant IV: ______________ and ________________ are positive

2.5 Reference Angles Name: ________________________________

Practice

Identify the quadrant in which the terminal side of θ lies.

1. sin θ>0 ,cot θ>0 2. sin θ<0 ,cosθ<0

Find the reference angle for each of the following. Then find sin θ for each angle.

3. 102° 4. 137°

5. 7π6

6. 5π3

7. 1090° 8. 478°

2.6 Trigonometric Functions Name: ________________________________

Obj.: I will be able to use the definition of trigonometric functions of any angle to evaluate trig functions for any angle. I will be able to determine the sign of a trig function given the quadrant in which lies the terminal side of the angle.

Notes Use the _________________ of trigonometric functions of _______ _____________

o In the examples below, the angles are in standard position, and _________ is ____ units from the origin ____ the ______________ side of θ.

o A _____________ triangle is formed by drawing the line segment from P ____________ to the x-axis. In this case, y is _______________ θ, and x is _______________ to θ.

o Definitions of Trigonometric Functions of Any Angle P (x,y) is any point on the __________________ side other than the origin. ___________________ (the distance from the origin to P) The six trigonometric ratios of θ are ______________ by the following ratios:

2.6 Trigonometric Functions Name: ________________________________

Practice

Find the exact value of each of the six trigonometric functions of θ if P(x,y) is on the terminal side of angle θ.

1. P(2,1) 2. P(4,5)

Evaluate the following trigonometric functions at the quadrantal angle, or state that the expression is undefined3. sin 0 4. cot π

Find two values of θ ,0≤θ≤2π , that satisfy the following equations.5.

sin θ=−√22

6.cosθ=−√3

2

Find the exact value of each of the remaining trigonometric functions of θ.7. cosθ=−5

13;θin Quadrant III

8. cosθ=2425,270 °<θ<360 °