Lesson QuizLesson Presentation

Lesson 9.1Use Trigonometry with

Right Triangles

Warm-Up

Standard Accessed: Students will prove, apply, and model trigonometric functions and ratios.

Warm-Up

2. 𝒄=𝟏𝟎 𝒂=√𝟓𝟏1.

In right triangle ABC, a and b are the lengths of the legs and c is the length of the hypotenuse. Find the missing length. Give exact values.

3. If you walk 2.0 kilometers due east and then 1.5 kilometers due north, how far will you be from your starting point.

𝟐 .𝟓𝑲𝒊𝒍𝒐𝒎𝒆𝒕𝒆𝒓𝒔1. distance formula

Vocabulary

Essential Understandings

How are trigonometric functions used in right triangles? The six trigonometric ratios: sine, cosine, tangent,

cosecant, secant, and cotangent, are the six possible ratios of pairs of sides of a right triangle.

If you know the length of any side and the measure of either of the acute angles, you can find all the remaining parts of a right triangle.

Evaluate trigonometric functionsEXAMPLE 1

SOLUTION

Evaluate the six trigonometric functions of the angle .

𝒔𝒊𝒏=𝟕𝟐𝟓

1.

𝜃24

7

2.

25

𝒄𝒐𝒔=𝟐𝟒𝟐𝟓

3.

𝒕𝒂𝒏=𝟕𝟐𝟒

𝒄𝒔𝒄=𝟐𝟓𝟕

4. 5.

𝒔𝒆𝒄=𝟐𝟓𝟐𝟒

6.

𝒄𝒐𝒕=𝟐𝟒𝟕

Evaluate trigonometric functionsEXAMPLE 2

SOLUTION

a.

If is an acute angle of a right triangle and , what is the value of ?

b.

c.

d.

Geometry Conjectures

In an isosceles right triangle, if the legs have length , then the hypotenuse has length .

In a triangle, if the shorter leg has length , then the longer leg has length and the hypotenuse has length .

Vocabulary

Find an unknown side length of a right triangleEXAMPLE 3

SOLUTION

Find the value for in the right triangle shown.

𝑥=3√2𝑜𝑟 4.243

cos 45 𝑖𝑠 √22

= 𝑥6

45 °𝑥6

Geometry Isosceles Conj.

𝑥=3√2𝑜𝑟 4.243

Use a calculator to solve a right triangleEXAMPLE 4

SOLUTION

Solve .

𝑝=11 .756

cos 54 °=¿𝑎𝑑𝑗 (𝑝 )20

¿

54 °

𝐺 𝑃

𝑅=20𝑝

𝑟

sin 54 °=¿𝑜𝑝𝑝 (𝑟 )20

¿

r=16 .180∠𝑃=36 °90 °+54 °+𝑃=180 °

Using indirect MeasurementEXAMPLE 4Hiking in Nepal You are hiking toward Machapuchare “Fish Tail” in the Annapurna range, but you reach a point where an avalanche has destroyed the trail (1). To avoid the avalanche, you take an alternative trail route. You turn onto a diagonal trail (2) that meets your original trail at a 48° angle and follow that trail for 3.6 miles until you hit another trail (3) that intersects back with your original trail at a 90° angle. How far were you from the intersection of your trail (1) and trail (3) when you turned onto the diagonal trail (2)? How far will you travel taking the alternative trail route around the avalanche?

≈ 2.4 mi48 °

𝑡𝑟𝑎𝑖𝑙(1)

3.6𝑚𝑖 𝑡𝑟𝑎𝑖𝑙(3)𝑡𝑟𝑎𝑖𝑙(2)

You will travel ≈ 6.3 mi around the avalanche.

Using Angle of ElevationEXAMPLE 5KIS Flagpole You measure from a point on the ground 28 feet from the base of the KIS flagpole, the angle of elevation to the top of the flagpole is 63°. Estimate the height of the flagpole.

The approximate height of the KIS flagpole is 55 feet.

SOLUTION

𝑥≈54.953

tan 63= 𝑥28

Lesson 9.1 Homework:Practice BPractice C “Honors”