6.1 Right-Triangle Trigonometry

Objectives:

1. Define the six trigonometric ratios of an acute angle in terms of a right triangle.

2. Evaluate trigonometric ratios, using triangles and on a calculator.

Units of Measure in a Circle

Degrees: One degree is 1/360th of a circle.

Minutes: One minute is 1/60th of a degree.

Seconds:

One second is 1/60th of a minute or 1/3600th of a degree.

Degrees are not the smallest unit of measure in a circle. Sometimes measurements are written with Degree, Minutes, & Seconds (DMS Form).

Ex. #1 Converting Between Decimal Form and DMS Form

A. Write in decimal form: "9 '30 42

5025.42

0025.05.0423600

9

60

3042

Since there are 60 seconds in a minute, the 9” needs divided by 60 twice, or just divided by 3600 which is 60(60).

Ex. #1 Converting Between Decimal Form and DMS Form

B. Write in DMS form: 4125.31

"45'2431

"6075.0'2431

'75.2431

'604125.031

Truncate the decimal

by removing whole units and multiply the remainder by seconds.

Repeat the process a second time and you have DMS Form.

Trigonometric Ratios

Remember Soh – Cah – Toa

)tan(

1)cot(

)cos(

1)sec(

)sin(

1)csc(

djacent

pposite)an(

ypotenuse

djacent)os(

ypotenuse

pposite)(in

a

ot

h

ac

h

os

Memorizing the Reciprocal Functions

The reciprocal functions can be memorized by remembering that the prefix of “co-” is used only once in each pair. Start with the easiest pair to remember:

tangent / cotangent sine / cosecant cosine / secant

Ex. #2 Evaluating Trigonometric Ratios

Evaluate the six trigonometric ratios of the angle θ, as shown below:

3

4tan

5

3cos

5

4sin

adj

opp

hyp

adj

hyp

opp

4

3

tan

1cot

3

5

cos

1sec

4

5

sin

1csc

opp

adj

adj

hyp

opp

hyp

Ex. #3 Evaluating Trig. Ratios on a Calculator

Evaluate the six trigonometric ratios of 15° using a calculator.NOTE: Make sure your calculator is set to Degree Mode first!

7321.315tan

115cot

0352.115cos

115sec

8637.315sin

115csc

2679.015tan

9659.015cos

2588.015sin

The 3 main functions are easy to enter, but to do the reciprocal functions we must do what their name says, take the reciprocal.

Ex. #4 Evaluating Trig. Ratios of Special Angles

Evaluate the six trigonometric ratios of 30°, 60°, & 45° on the triangles below:

Ex. #4 Evaluating Trig. Ratios of Special Angles

Evaluate the six trigonometric ratios of 30°, 60°, & 45° on the triangles below:

3

3

3

1

330tan

2

3

2

330cos

2

1

230sin

x

x

adj

opp

x

x

hyp

adj

x

x

hyp

opp

31

330cot

3

32

3

230sec

11

230csc

Finding the reciprocal functions on this is fairly easy. Some values may still need rationalized.

Ex. #4 Evaluating Trig. Ratios of Special Angles

Evaluate the six trigonometric ratios of 30°, 60°, & 45° on the triangles below:

33

60tan

2

1

260cos

2

3

2

360sin

x

x

adj

opp

x

x

hyp

adj

x

x

hyp

opp

3

3

3

160cot

11

260sec

3

32

3

260csc

For 60° the values for sine and cosine switch places as well as the values for tangent and cotangent.

Ex. #4 Evaluating Trig. Ratios of Special Angles

Evaluate the six trigonometric ratios of 30°, 60°, & 45° on the triangles below:

11

145tan

2

2

2

1

245cos

2

2

2

1

245sin

x

x

adj

opp

x

x

hyp

adj

x

x

hyp

opp

130cot

21

230sec

21

230csc

For 45° sine and cosine have the same values.