Sec. 8 – 3 Sec. 8 – 3 The Tangent The Tangent

RatioRatio

Objective:Objective:

1) To use tangent ratios 1) To use tangent ratios to to determine side determine side lengths in lengths in ΔΔ..

This only for right This only for right ΔΔs!!s!!

TrigonometryTrigonometry Greek WordGreek Word Trigon → TriangleTrigon → Triangle Metron → MeasureMetron → Measure

Trigonometry RatioTrigonometry Ratio – Ratio of the – Ratio of the lengths of sides of a right lengths of sides of a right ΔΔ..

The tangent is just a button on your The tangent is just a button on your calculator!calculator!

TaTann** Make sure you calculator is in

Degrees!!

Tangent RatioTangent Ratio

Tangent RatioTangent Ratio – Ratio of the length – Ratio of the length of the opposite leg from an of the opposite leg from an to the to the length of the leg adjacent to the length of the leg adjacent to the same same ..A

C B

b

a

c

* Can’t use the right , CTangent A =

Length of leg Opposite of ALength of leg Adjacent of A

Tangent A =

ab

Writing tangent ratiosWriting tangent ratios

Write the tangent ratio of Write the tangent ratio of T and T and U.U.

T

U

S

6

8

10

Tangent =OppositeAdjacent

Tangent T = 8

6TSUS

= Tangent U =

TSUS 6

8=

** Tangent ratio for T & U are reciprocals

You can use the tangent You can use the tangent ratio to find the measure of ratio to find the measure of a distance that is difficult to a distance that is difficult to

measure directly.measure directly. Example 1: Find w.Example 1: Find w.

10

w

54

Step 1: Set up the Tangent RatioTan 54 =opp

adj

Tan 54 =w10

1.376 =w10

13.76 = w

Ex. 2: Solve for the variable Ex. 2: Solve for the variable using the tangent ratio.using the tangent ratio.

70°

8cm

x

Step 1: Set up the tangent ratio.

Tan 70 =oppadj

Tan 70 =8x

2.747 = 8x Multiply both sides

by the denominator, x2.747x = 8

x = 2.9

The Tangent Inverse: The Tangent Inverse: TanTan-1-1

Just another button on your Just another button on your Calculator!Calculator!

Use it when you have the two sides Use it when you have the two sides of a of a ΔΔ and are trying to find a and are trying to find a missing missing ..

TanTanTan-1Use the SHIFT

(2nd) Key to get to it.

Once you press it, it should look like this:Tan-1 (

Ex.3: Using the TanEx.3: Using the Tan-1-1

Use the TanUse the Tan-1-1 to solve for the missing to solve for the missing .. 12m

m5mm

y°

Step 1: Set up the Tan Ratio

Tan y = oppadj

Tan y =512

Tan y = .4167

At this point you will use the Tan-1:

1)Hit shift Tan to get to

Tan-1(

2) Type in the decimal and hit enter

Tan-1 (.4167) =

22.6°

Ex.4: Solve for mEx.4: Solve for mZZ

8mil8mileses

6

6

mile

sm

iles

x Y

ZTan Z = opp

adj

Tan Z = 86

Tan Z = 1.333

Tan-1 (1.333) = mZ

mZ = 53.1°

What have I learned???What have I learned???

Tan Tan = =

Use TanUse Tan-1-1 when looking for an when looking for an measure.measure.

Opposite SideAdjacent Side