Trigonometric Ratios Success Criteria: I can solve problems involving right triangles using trig ratios I can find sin, cos, tangent of an acute angle

Trigonometric Ratios

Success Criteria: I can solve problems involving

right triangles using trig ratios I can find sin, cos, tangent of

an acute angle in a right triangle

Today’s Agenda Entry task Homework Check Lesson Notebook Check Happy Spring Break

8.3 Trigonometry (Day 1)Understand what trigonometric ratios are and

how to use themDo Now – You Need a Calculator!!Write the fraction as a decimal rounded to the nearest hundredth.

1.

Solve each equation.

2.

0.29

x = 7.25



Trigonometric RatiosDo Not Copy – Listen!!

By the AA Similarity Postulate, a right triangle with a given acute angle is similar to every other right triangle with that same acute angle measure.

So ∆ABC ~ ∆DEF ~ ∆XYZ, and . These are trigonometric ratios.

A trigonometric ratio is a ratio of two sides of a right triangle.



It is pronounced "so - ka - toe - ah".

The SOH stands for "Sine of an angle is Opposite over Hypotenuse."

The CAH stands for "Cosine of an angle is Adjacent over Hypotenuse."

The TOA stands for "Tangent of an angle is Opposite over Adjacent."

The Legend of SOH – CAH - TOA


In trigonometry, the letter of the vertex of the angle is often used to represent the measure of that angle. For example, the sine of A is written as sin A.

Writing Math


Example 1

Write the trigonometric ratio as a fraction and as a decimal rounded tothe nearest hundredth.

sin B


cos J

Example 2: Finding Trigonometric Ratios

Write the trigonometric ratio as a fraction and as a decimal rounded to the nearest hundredth.


tan K

Example 3: Finding Trigonometric Ratios

Write the trigonometric ratio as a fraction and as a decimal rounded to the nearest hundredth.


Calculating Trigonometric Ratios

Use your calculator to find the trigonometric ratio. Round to the nearest hundredth.

sin 52°

sin 52° 0.79

Be sure your calculator is in degree mode, not radian mode.

Caution!


Calculating Trigonometric Ratios

Use your calculator to find the trigonometric ratio. Round to the nearest hundredth.

cos 19°

cos 19° 0.95


BC 38.07 ft

Write a trigonometric ratio.

Substitute the given values.

Multiply both sides by BC and divide by tan 15°.

Simplify the expression.

Example 4: Using Trigonometric Ratios to Find Lengths

Find the length. Round to the nearest hundredth.

BC is adjacent to the given angle, B. You are given AC, which is opposite B. Since the adjacent and opposite legs are involved, use a tangent ratio.


Do not round until the final step of your answer. Use the values of the trigonometric ratios provided by your calculator.

Caution!


Write a trigonometric ratio.

Substitute the given values.

12.9(sin 63°) = QR

11.49 cm QR

Multiply both sides by 12.9.

Simplify the expression.

Example 5: Using Trigonometric Ratios to Find Lengths

Find the length. Round to the nearest hundredth.

QR is opposite to the given angle, P. You are given PR, which is the hypotenuse. Since the opposite side and hypotenuse are involved, use a sine ratio.





Today’s Agenda

Do Now:

No calculator draw the triangles

8.3 TrigonometryUnderstand what trigonometric ratios are and

how to use them

Use a special right triangle to write each trigonometric ratio as a fraction.

1. sin 60° 2. cos 45°


Example 1: Solving Right Triangles

Find the unknown measures. Round angle measures to the nearest degree.

Since the acute angles of a right triangle are complementary, mT 90° – 29° 61°.

Trigonometric RatiosExample 1: Discuss with your table:

Now that you know the angle measures, how could you find the length of ST?

Find the unknown measures. Round angle measures to the nearest degree.

Since the acute angles of a right triangle are complementary, mT 90° – 29° 61°.

Trigonometric RatiosIf you know the sine, cosine, or tangent of an acute angle measure, you can use the inverse trigonometric functions to find the measure of the angle.


Example 2: Calculating Angle Measures from Trigonometric Ratios

Use your calculator to find each angle measure to the nearest degree.

A. cos-1(0.87) B. sin-1(0.85) C. tan-1(0.71)

cos-1(0.87) 30° sin-1(0.85) 58° tan-1(0.71) 35°

Trigonometric RatiosExample 2

Find the unknown measures. Round lengths to the nearest hundredth and angle measures to the nearest degree.

Since the acute angles of a right triangle are complementary, mD = 90° – 58° = 32°.

, so EF = 14 tan 32°. EF 8.75

DF2 = 142 + 8.752

DF2 = ED2 + EF2

DF 16.51

Trigonometric RatiosExample 3: Identifying Angles from Trigonometric Ratios

Use the trigonometric

ratio to determine the

measure of 1 and 2 .


Example 4

Use the trigonometric

ratio to determine the

measure of 1 and 2 .

Trigonometric RatiosAssignment #30 pg 510

#11-19, 22-24





Today’s Agenda

Do Now:

8.3 TrigonometryUnderstand what trigonometric ratios are and

how to use them

Use your calculator to find each trigonometric ratio. Round to the nearest hundredth.

1. tan 84° 2. cos 13°

9.51 0.97


Lesson Quiz: Part II

Find each length. Round to the nearest tenth.

5. CB

6. AC

6.1

16.2

Use your answers from Items 5 and 6 to write each trigonometric ratio as a fraction and as a decimal rounded to the nearest hundredth.

7. sin A 8. cos A 9. tan A


Example 2: Finding Trigonometric Ratios in Special Right Triangles

Use a special right triangle to write cos 30° as a fraction.

Draw and label a 30º-60º-90º ∆.


Check It Out! Example 2

Use a special right triangle to write tan 45° as a fraction.

Draw and label a 45º-45º-90º ∆.

s

45°

45°

s


Do Now

Find each length. Round to the nearest tenth.

1. CB

2. AC

6.1

16.2

Documents

Trigonometric Ratios Success Criteria: I can solve problems involving right triangles using trig ratios I can find sin, cos, tangent of an acute angle