Math 10

Ms. Albarico

Students are expected to:•Demonstrate an understanding of and apply properties to operations involving square roots.

• Relate the trigonometric functions to the ratios in similar right triangles.

• Use calculators to find trigonometric values of angles and angles to find when trigonometric values are known.

* Solve problems using the trigonometric ratios.

A RATIO is a comparison of two numbers. For

example; boys to girls cats : dogs

right : wrong.

In Trigonometry, the comparison is between

sides of a triangle.

We need to do some housekeeping before we can proceed…

In trigonometry, the ratio we are talking about is the comparison of the sides of a RIGHT TRIANGLE.

Two things MUST BE understood:1. This is the hypotenuse.. This will

ALWAYS be the hypotenuse2. This is 90°… this makes the right

triangle a right triangle…. Without it, we can not do this trig… we WILL NOT use it in our calculations because we COULD NOT do calculations without it.

Now that we agree about the hypotenuse and right angle, there are only 4 things left; the 2 other angles and the 2 other sides.A We will refer to the sides

in terms of their proximity to the angle

If we look at angle A, there is one side that is adjacent to it and the other side is opposite from it, and of course we have the hypotenuse.

opposite

adjacenthypotenuse

B

If we look at angle B, there is one side that is adjacent to it and the other side is opposite from it, and of course we have the hypotenuse.

opposite

adjacent

hypotenuse

Remember we won’t use the right angle

X

One more thing…

Here we go!!!!

Trigonometric Ratios

Name“say” Sine Cosine tangent

AbbreviationAbbrev. Sin Cos Tan

Ratio of an angle measure

Sinθ = opposite side hypotenuse

cosθ = adjacent side hypotenuse

tanθ =opposite side adjacent side

Values of Trigonometric Function

00 300 450 600 900

Sine 0 0.5 1/2 3/2 1

Cosine 1 3/2 1/2 0.5 0

Tangent 0 1/ 3 1 3 Not defined

Cosecant Not defined 2 2 2/ 3 1

Secant 1 2/ 3 2 2 Not defined

Cotangent Not defined 3 1 1/ 3 0

One more time…Here are the ratios:

sinθ = opposite side hypotenuse

cosθ = adjacent side hypotenuse

tanθ = opposite side adjacent side

B c a

C b A

Write the ratio for sin A

Sin A = a c

Write the ratio for cos A

Cos A = b c

Write the ratio for tan A

Tan A = a b

Let’s switch angles: Find the sin, cos and tan for Angle B:

Sin B = b

cCos B = a

c

Tan B = b

a

Set your calculator to ‘Degree’…..

MODE (next to 2nd button)

Degree (third line down… highlight it)

2nd

Quit

Calculator Commands

Find tan A: 24.19 12

A 21

Tan A = opp/adj = 12/21

Tan A = .5714

8

4A

Tan A = 8/4 = 2 8

Find tan A:

Note:

Given Looking for Use

Ratio of sidesAngle measure

SIN-1

COS-1

TAN-1

Angle, side Missing side SIN, COS, TAN

Set your calculator to ‘Degree’…..

MODE (next to 2nd button)

Degree (third line down… highlight it)

2nd

Quit

Calculator Commands Reminder

To solve for Angles:

A B

C

xo

Opp’

Adj’

adjoppxo tan

Now we need to look at the two ratios involving the hypotenuse:

sin xo = OppositeHypotenuse

hyp’

cos xo = AdjacentHypotenuse

Calculator Commands

For Trigonometric Inverse Functions:

1) Press 2ND, useSIN for SIN-1

COS for COS-1

TAN for TAN-1

Calculate the angle b o below.

14.8cm

9.7cmb o

(1) Identify the two sides marked.a

h

(2) Choose the correct trig ratio .

(3) Substitute in values .

(4) Calculate the ratio(3 decimal places).

(5) Use the inverse cosine function on your calculator to calculate the angle .b o = 49.1o

hypadjxo cos

8.147.9cos ob

655.0cos ob

655.0cos 1ob

Remembering the Trigonometric Ratios:Look again at the three trig ratios given below:

hypoppxo sin

hypadjxo cos

adjoppxo tan

Take the first letter of each word.

Write the letters in order.

S O H C A H T O A

C2cm

B 3cm A

Find an angle that has a tangent (ratio) of 2

3

Round your answer to the nearest degree.

Process:

I want to find an ANGLE.

I was given the sides (ratio).

Tangent is opp

adj

Solution:

TAN-1(2/3) = 34°

Ok… we’ve found side lengths, now let’s find angle measures.

Refer to your table… what function will we use to find angle measures?

SIN-1COS-1TAN-1 These are

called INVERSE FUNCTIONS.

Homework!

In your notebook, CYU # 18, 19, 20, 21, 22, 24, and 25 on pages 239-240.

Class Work

In your notebook, solve the following:

CYU # 12, 13, 14, 15, 16 on pages 236-237.

Work Period

Work with your group members about the final design of your pet house.

Remember :

Your scale drawing must be accurate and precise.