# Angles, Arc, and Measure

Angles and Measure

Angles - An angle is formed by rotating a ray about its endpoint from some _____________ position to some ___________________ position.

Standard Position - An angle with its ______________ at the __________ and the __________________________ along the ____________________ is in standard position.

Positive Angles - An angle measured ______________________________ is positive.

Negative Angles - An angle measured ___________________________ is negative.

Degrees

1 revolution =

½ revolution =

¼ revolution =

1 revolution clockwise =

2 revolutions =

Minutes - Each degree is divided into _____ equal parts. Each is called a ________________.

Seconds - Each minute is divided into _____ equal parts. Each is called a _____________. There are _________ seconds in a degree.

Converting between Minutes and Seconds and Degree Decimals

27o 35' 15" 130o 4' 50"

Radians

1 revolution =

½ revolution =

¼ revolution =

1 revolution clockwise =

2 revolutions =

Converting between Radians and Degrees π =

Convert Degrees to Radians Examples:

135o 540o -270o

Convert Radians to Degrees Examples:

[pic] [pic] [pic]2

How many degrees is one radian?

Coterminal Angles - Angles in standard position that have the same _________________________________ are called coterminal angles.

Are 280o and 1000o coterminal? Are they equal?

Example: Find four coterminal angles to 60o.

Make at least one of them negative!

(Add or subtract ________________ or ____________)

Complementary Angles - Two angles are complementary when their sum is _____________.

Supplementary Angles - Two angles are supplementary when their sum is _______________.

Examples: Find the complementary and supplementary angles.

35o

178o

[pic]

[pic]

Triangle Definition of Trig Functions

sin [pic] = csc [pic] =

cos [pic] = sec [pic] =

tan [pic] = cot [pic] =

Special Triangles

sin 30o = sin 60o = sin 45o =

cos 30o = cos 60o = cos 45o =

tan 30o = tan 60o = tan 45o =

Example: Find the exact values of the six trigonometric functions of the angle θ given in the figure.

Example: Sketch a right triangle corresponding to the trigonometric function of the acute angle θ. Find the other five trig functions of θ.

Example: Use a calculator to evaluate each function. Round your answers to four decimal places. Be sure your calculator is in the correct mode.

[pic]= [pic]=

[pic]= [pic]=

[pic]= [pic]=

[pic]= [pic]= [pic]=

Example: Solve for the variable.

Example: A surveyor is standing 50 feet from the base of a large tree. The surveyor measures the angle of elevation to the top of the tree as 71.5o. How tall is the tree?

Ratio Definition of Trig Functions

Draw a point (x, y) on the terminal side of the angle.

Make a "reference triangle."

sin θ = csc θ =

cos θ = sec θ =

tan θ = cot θ =

What if we know a point on the terminal side of the angle but not the measure of the angle? Find all six trig functions.

We are not limited to just the first quadrant.

Find all six trig functions.

Note: r is always __________

Find sin, cos, & tan for all four angles.

These trig values are the same for ____________________angles.

Examples: Sketch θ in standard position, and then determine its reference angle.

A. 203o B. 127o

C. -350o D. 315o

E. [pic] F. [pic]

G. [pic] H. [pic]

Example: Evaluate the sine, cosine, and tangent of the angles without a calculator.

A. 300o B. 120o

C. -120o D. -315o

Example: Evaluate the sine, cosine, and tangent of the angles without a calculator.

E. [pic] F. [pic]

G. [pic] H. [pic]

Examples: Find two values of θ that satisfy the equation. Give your answers in BOTH degrees and radians.

A. [pic] B. [pic]

C. [pic] D. [pic]

Examples:

If [pic] in QIV, what does [pic] If [pic] in QIII, what does [pic]

Inverse Trig Functions

We have done problems like this:

We have done problems like this:

The ratio is the _____________________ between two sides of a ____________.

We have done problems like this:

What about problems like this?:

Since this is not a triangle we know, we are limited to approximate answers using the calculator.

Do we want our angle answers in degrees or radians?

Draw a sketch. Find the _____________________________________.

Use the inverse button.

Take care with _________________.

Use __________________________.

There are 2 primary solutions, but why are there really an infinite number of solutions?

Example: A 12m flagpole casts a 9m shadow. Find θ, the angle of elevation of the sun.

How many answers are we looking for? Why?

Example:

Find values of that satisfy the equation.

Give your answers in degrees (0o ................

................

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