for each angle in standard position, determine 4 notes 2015.… · given any angle θ, we can use...

Chapter 4 Notes Trigonometry and the Unit Circle

1

Math 301

4.1 Angles and Angle Measure

Match the angle with its measure (in degrees).

a) 700 b) 1350 c) 3000 d) 700 e) 2200

Match the angle with its measure (in degrees).

a) 4300 b) 2250 c) 10200 d) 7900 e) 5800

coterminal angles: angles in standard position with the same terminal arms


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Math 301

Definition: One radian is the measure of the central angle subtended in a circle by an arc equal in length to the radius of the circle.

degrees radians

one full rotation:

one half rotation:

one quarter rotation:

one eighth rotation:

Note: Angle measures without units are assumed to be in radians.

Example:

For each angle in standard position, determine one positive and one negative angle measure that is coterminal with it.

a) 2700 b) 5π/4 c) 7400

The General Form

Any given angle has an infinite number of angles coterminal with it. Angles coterminal with any angle θ can be expressed as:


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Math 301

Unit ConversionsGiven any angle θ, we can use ratios to convert from degree measure to radian measure, and from radian measure to degree measure.

1 full rotation = degrees 1 full rotation = radians

Therefore, ___________________________________.

So, to convert from radians to degrees, we can multiply by

OR

and to convert from degrees to radians we can multiply by

OR

Example: Change each degree measure to radian measure and each radian measure to degree measure.

a) 300 b) 1200

c) 5π/4 d) 2.57


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Math 301

Arc Length of a Circle

a =

Note: This formula works for any circle, provided that is measured in radians, and a and r are both measured in the same units.


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Math 301

Summary: • angles can be measured using different units, including degrees and radians

• an angle measured in one unit can be converted to the other using the relationships 1 full rotation = 3600 = 2π

• angles that are coterminal have the same initial arm and the same terminal arm

• an angle has an infinite number of coterminal angles

Homeworkpage 175179 # 3, 4, 6, 9, 11, 12, 14, 19, C1

Push Yourself! Try #27


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Math 301

The Equation of the Unit Circle

The equation of the unit circle can be used to check and/or find coordinate points along the unit circle.

4.2 The Unit Circle

Ex) Determine the coordinates for all points on the unit circle where the xcoordinate is 2/3.

Ex) Determine the coordinates for all points on the unit circle where the ycoordinate is 1/√2 and the point is in quadrant 3.


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Math 301

The Unit Circle:

Key Points• The equation for the unit circle is x2 + y2 = 1. It can be used to determine whether a point is on the unit circle or to determine the value of one coordinate given the other.

• On the unit circle, the measure in radians of the central angle and the arc subtended by that central angle are numerically equivalent.

• Some of the points on the unit circle correspond to exact values of the special angles learned previously.

Practice

page 186189 # 15, 1012, 15, 17


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Math 301

4.3 Trigonometric Ratios

Primary Trigonometric Ratios:

Reciprocal Trigonometric Ratios:

Recall: CAST


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Math 301

Reference AnglesFor any given angle, its reference angle is an acute version of that angle. In standard position, the reference angle is the smallest angle between the terminal arm and the xaxis.

The values of the trig functions of angle θ are the same as the trig values of the reference angle for θ, give or take a minus sign.

Ex

ExamplePoint A (12/13, 5/13) is on the unit circle and on the terminal arm of an angle θ in standard position. Determine the values for the six trigonometric ratios for angle θ.


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Math 301Ex

Example

Consider the point P(2, 6). Determine the values for the six trigonometric ratios for angle θ. Round your answers to the nearest tenth.

Exact Values of Trigonometric RatiosYou can determine the exact value for a trigonometric ratio using the special triangles you learned about in Math 20.

Example

Determine the exact value for each ratio.


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Math 301

Approximate Values of Trigonometric Ratiosin degree mode in radian mode

sin(30)

sin(π/6)

cos(135)

cos(3π/4)

You can determine approximate values for trigonometric ratios using technology, provided that your calculator is

in the right mode!

ExampleDetermine the approximate value for each trigonometric ratio. Give your answers to four decimal places.

tan cos 2600

sin 4.2 csc (700)

7π5


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Math 301

Key Ideas• P(θ) = (cosθ, sinθ)• the reciprocal ratio of sine is cosecant• the reciprocal ratio of cosine is secant• the reciprocal ratio of tangent is cotangent• exact values of trig ratios for special angles and their multiples may be determined using the coordinates of points on the unit circle

• you can determine approximate values for trigonometric ratios using a calculator in the appropriate mode!!

Homework

pg 201203 # 16 (odds), 8, 9, 13, 16, 17Push Yourself! Try # 20 and 22!

4.4 Introduction to Trigonometric EquationsMany of the strategies for solving trigonometric equations mirror the strategies for solving linear and quadratic equations.

An understanding of coterminal angles, points on the unit circle, and inverse trigonometric functions will be important for understanding the solution of trigonometric equations.


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Math 301

ExampleFind θ if cosθ = 0.8090 and 0 ≤ θ ≤ π.

ExampleFind θ if 4cotθ = 1 and 0O ≤ θ ≤ 180O.

ExampleFind θ if 4cscθ = 25 and 0 ≤ θ ≤ 2π.

ExampleSolve 2sinxcosx cosx = 0 for 0 ≤ x ≤ 2π.

What values satisfy x where 2sin2x = sinx for 0 ≤ x ≤ 2π ?


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Math 301

ExampleSolve 2sin2x + 3sinx 2 = 0 for 0 ≤ x ≤ 2π.

What values satisfy θ where sin2θ 1 = 0 for 0 ≤ θ ≤ 2π ?

General SolutionsA general solution is needed when the domain restrictions for a trigonometric equation are lifted (ie. θ ∈ R), because we often get an infinite number of solutions.

Use the following procedure to find general solutions using an algebraic approach:

1. Solve the equation for one rotation of the unit circle (ie. 0 ≤ θ ≤ 2π)

2. The general solution can be found the same way that we found coterminal angles (adding/subtracting 2π)


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Math 301

ExampleFind the general solution for each of the following.

2cosx √3 = 0 , x ∈ R

cotx = √3 , x ∈ R

3secx 6 = 0, x ∈ R

Summary• To solve a trigonometric equation algebraically, you can use the same technique as used in solving linear and quadratic equations

• When you arrive at sinθ = a or cosθ = a or tanθ = a, where a ∈ R, then use the unit circle for exact values of θ and inverse trigonometric function keys on a calculator for approximate measures. Use reference angles to find solutions in other quadrants.

• To solve a trigonometric equation involving cscθ, secθ, or cotθ, you may need to work with the related reciprocal value(s).

• To determine a general solution or if the domain is real numbers, find the solutions in one positive rotation (2π or 360O). Then, use the concept of coterminal angles to write an expression that identifies all possible measures.

Homeworkpage

3 5 (odds), 7, 9, 15 19

for each angle in standard position, determine 4 notes 2015.… · given any angle θ, we can use...

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