trigonometric ratios

TrigonometriTrigonometric Ratiosc Ratios

Please view this tutorial and Please view this tutorial and answer the follow-up questions answer the follow-up questions on loose leaf to turn in to your on loose leaf to turn in to your

teacher.teacher.

Identifying Parts of a Identifying Parts of a Right TriangleRight Triangle

Hypotenuse – always Hypotenuse – always across from the 90° angleacross from the 90° angle

Side Opposite – always Side Opposite – always across from the angle across from the angle being referencedbeing referenced

Side Adjacent- always Side Adjacent- always touching the angle being touching the angle being referencedreferenced

*Note that all angles are *Note that all angles are marked with capitol marked with capitol letters and sides are letters and sides are marked with lower case marked with lower case lettersletters

A

BC

Angle C measures 90°

Identifying Parts of a Identifying Parts of a Right TriangleRight Triangle

What side is opposite of What side is opposite of angle A?angle A? Side BCSide BC

What side is opposite of What side is opposite of angle B?angle B? Side ACSide AC

What side is adjacent to What side is adjacent to angle A?angle A? Side ACSide AC

What side is adjacent to What side is adjacent to angle B?angle B? Side BCSide BC

What side is the What side is the hypotenuse?hypotenuse? Side ABSide AB

A

BC

Trigonometric Ratios Trigonometric Ratios (only apply to right (only apply to right

triangles)triangles) Sine (abbreviated sin)Sine (abbreviated sin) Sin x° = Sin x° =

Example:Example:oppositehypotenuse

A

C B

Sin A = BCAB


triangles)triangles) Cosine (abbreviated Cosine (abbreviated

cos)cos) Cos x° = Cos x° =

Example:Example:adjacenthypotenuse

A

C B

Cos A = ACAB


triangles)triangles) Tangent (abbreviated Tangent (abbreviated

tan)tan) Tan x° = Tan x° =

Example:Example:oppositeadjacent

A

C B

Tan A = BCAC

Helpful Hint to Helpful Hint to Remember the Trig RatiosRemember the Trig Ratios SOH (sine = opposite / hypotenuse)SOH (sine = opposite / hypotenuse) CAH (cosine = adjacent / hypotenuse)CAH (cosine = adjacent / hypotenuse) TOA (tangent = opposite / adjacent)TOA (tangent = opposite / adjacent)

Remember SOH CAH TOARemember SOH CAH TOA

Time to PracticeTime to Practice

Identify the following trig ratio valuesIdentify the following trig ratio valuesC B

A

Sin A = Sin B=

Cos A = Cos B=

Tan A = Tan B=

3

4 5

Time to PracticeTime to Practice

Identify the following trig ratio valuesIdentify the following trig ratio valuesC B

A

Sin A = Sin B=

Cos A = Cos B=

Tan A = Tan B=

3

4 5

35

45

34

45

35

43

More PracticeMore Practice

Identify the following trig ratio valuesIdentify the following trig ratio values

C

B

A

Sin A = Sin B=

Cos A = Cos B=

Tan A = Tan B=12

13 5

More PracticeMore Practice

Identify the following trig ratio valuesIdentify the following trig ratio values

C

B

A

Sin A = Sin B=

Cos A = Cos B=

Tan A = Tan B=

12

13 5

513

512

1213

513

125

1213

How to use the trig ratios How to use the trig ratios to find missing sidesto find missing sides

Step 1: Make sure your calculator is in Step 1: Make sure your calculator is in degree modedegree mode

Step 2: Label the right triangle with the Step 2: Label the right triangle with the words opposite, adjacent, and hypotenuse words opposite, adjacent, and hypotenuse based on the given angle based on the given angle (Note: Do not use (Note: Do not use the right angle.)the right angle.)

Step 3: From the given information, Step 3: From the given information, determine which trig ratio should be used determine which trig ratio should be used to find the side length to find the side length

Step 4: Substitute in the given informationStep 4: Substitute in the given information

How to use the trig ratios How to use the trig ratios to find missing sides to find missing sides

(continued)(continued) Step 5: Put a 1 under the trig ratioStep 5: Put a 1 under the trig ratio Step 6: Cross multiplyStep 6: Cross multiply Step 7: When x=, put problem into your Step 7: When x=, put problem into your

calculator (Note: you may have to divide first calculator (Note: you may have to divide first to get x by itself) to get x by itself)

(NOTE: The angles of a triangle MUST add up (NOTE: The angles of a triangle MUST add up to be 180°)to be 180°)

ExampleExample

Given the following triangle, solve for x.Given the following triangle, solve for x.

8 cm

x

60°

Let’s Talk Through the Let’s Talk Through the StepsSteps

Step 1 : Check calculator for degree modeStep 1 : Check calculator for degree mode

Press the Mode button and make sure Press the Mode button and make sure Degree is highlighted as in the picture Degree is highlighted as in the picture belowbelow

Step 2Step 2

Label the triangle according to the given Label the triangle according to the given angleangle

8 cm- HYPOTENUSE

X - OPPOSITE

60°

Step 3Step 3

Identify the trig ratio we should use to solve Identify the trig ratio we should use to solve for x.for x.

8 cm- HYPOTENUSE

X - OPPOSITE

60°

From the 60° angle, we know the hypotenuse and need to find the opposite. So we need to use SINE.

Step 4Step 4

Substitute in the given information into the Substitute in the given information into the equation.equation.

8 cm- HYPOTENUSE

X - OPPOSITE

60°Sin x°=

Sin 60° =

oppositehypotenuse

x8

Step 5Step 5

Put a 1 under the trig ratioPut a 1 under the trig ratio

8 cm- HYPOTENUSE

X - OPPOSITE

60°Sin x°=

Sin 60° = 1

oppositehypotenuse

x8

Step 6Step 6

Cross multiply to solve for xCross multiply to solve for xSin 60° = 1

x8

8 sin (60°) = x

Step 7Step 7

Since x is already by itself, I can enter the Since x is already by itself, I can enter the information into the calculator.information into the calculator.

Therefore, we can state that x=6.93.

Let’s Look at Another Let’s Look at Another ExampleExample

Suppose that when we set-up the ratio Suppose that when we set-up the ratio equation, we have the following:equation, we have the following:

Tan 20° = Tan 20° = 4x

What Happens When We What Happens When We Cross Multiply?Cross Multiply?

Tan 20° Tan 20° = =

11

X tan 20° = 4X tan 20° = 4 (How do we get x by itself?) (How do we get x by itself?)

tan 20° tan 20° (Now we have to divide by tan 20° in order tan 20° tan 20° (Now we have to divide by tan 20° in order to solve for x)to solve for x)

X =X = 44

tan 20°tan 20°

X = 10.99X = 10.99

4x

How to use the trig ratios How to use the trig ratios to find missing anglesto find missing angles

Step 1: Make sure your calculator is in Step 1: Make sure your calculator is in degree mode (See slide 15)degree mode (See slide 15)

Step 2: Label the right triangle with the Step 2: Label the right triangle with the words opposite, adjacent, and hypotenuse words opposite, adjacent, and hypotenuse based on the given angle based on the given angle (Note: Do not use (Note: Do not use the right angle.)the right angle.)

Step 3: From the given information, Step 3: From the given information, determine which trig ratio should be used determine which trig ratio should be used to find the side length to find the side length

Step 4: Substitute in the given informationStep 4: Substitute in the given information

How to use the trig ratios How to use the trig ratios to find missing sides to find missing sides

(continued)(continued) Step 5: Solve for x by taking the inverse Step 5: Solve for x by taking the inverse

(opposite operation) of the trig ratio.(opposite operation) of the trig ratio. Step 6: When x=, put problem into your Step 6: When x=, put problem into your

calculator.calculator.

Calculator Steps for Calculator Steps for Finding AnglesFinding Angles

To solve for x, remember to take the inverse To solve for x, remember to take the inverse trig function.trig function.

On the calculator, you can find the inverse On the calculator, you can find the inverse trig functions by pressing 2trig functions by pressing 2ndnd and then the and then the trig function.trig function.

sin−1

cos−1

tan−1

Let’s Look at an Let’s Look at an ExampleExample

Given the following triangle, solve for x.Given the following triangle, solve for x.

200 cm

62 cm90°

x

Step 2Step 2

Label the sides opposite, adjacent, or Label the sides opposite, adjacent, or hypotenuse from angle x.hypotenuse from angle x.

200 cm

62 cm90°

x

OPPOSITE

HYPOTENUSE

Step 3Step 3

Since we have the opposite and the Since we have the opposite and the hypotenuse, we need to use SINE.hypotenuse, we need to use SINE.

200 cm

62 cm90°

xHYPOTENUSE

OPPOSITE

Step 4Step 4

Substitute in the given information into the Substitute in the given information into the equation.equation.

200 cm

62 cm90°

xHYPOTENUSE

OPPOSITESin x = 62

200

Step 5Step 5

To solve for x, we need to take the inverse of To solve for x, we need to take the inverse of sine on both sides.sine on both sides.

200 cm

62 cm90°

xHYPOTENUSE

OPPOSITESin x = 62

200

Sin-1 (sin x) = Sin-1 62200

⎛⎝⎜

⎞⎠⎟

Step 6Step 6

Now just type in the x= on your calculator.Now just type in the x= on your calculator.Sin x = 62

200Sin-1 (sin x) = Sin-1 62

200⎛⎝⎜

⎞⎠⎟

X = Sin-1 62200

⎛⎝⎜

⎞⎠⎟

X = 18°

Now It’s Your Turn!Now It’s Your Turn!

Use what you’ve just reviewed to help you Use what you’ve just reviewed to help you answer the following questions.answer the following questions.

Submit all of your work to your teacher Submit all of your work to your teacher after completing the tutorial.after completing the tutorial.

Don’t be afraid to go back through the Don’t be afraid to go back through the slides if you get stuck.slides if you get stuck.

GOOD LUCK!GOOD LUCK!

Problem #1Problem #1

Complete the following ratios.Complete the following ratios.

C A

B

90°6 cm

8 cm10 cm

Sin A = Sin B =

Cos A= Cos B=

Tan A= Tan B=


Solve for x and y.Solve for x and y.

90°

55°

x

40 fty


Solve for angles A and B.Solve for angles A and B.

90°

A

5 inB

C7 in

trigonometric ratios

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