1. Lesson 9.1 Use Trigonometry with Right Triangles
Standard Accessed: Students will prove, apply, and model trigonometric functions and ratios.
Warm-Up
Lesson Presentation
Lesson Quiz

2. Warm-Up
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.
1. 𝐚=𝟔, 𝐛=𝟖
2. 𝒄=𝟏𝟎, 𝒃=𝟕
𝒄=𝟏𝟎
𝒂= 𝟓𝟏
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.
1. 𝐝= 𝒙 𝟐 − 𝒙 𝟏 𝟐 + 𝒚 𝟐 − 𝒚 𝟏 𝟐 distance formula
𝟐.𝟓 𝑲𝒊𝒍𝒐𝒎𝒆𝒕𝒆𝒓𝒔

3. Vocabulary

4. 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.

5. 1 25 𝒔𝒊𝒏= 𝟕 𝟐𝟓 𝒄𝒐𝒔= 𝟐𝟒 𝟐𝟓 𝒕𝒂𝒏= 𝟕 𝟐𝟒 𝒄𝒔𝒄= 𝟐𝟓 𝟕 𝒔𝒆𝒄= 𝟐𝟓 𝟐𝟒 𝒄𝒐𝒕= 𝟐𝟒 𝟕
EXAMPLE 1
Evaluate trigonometric functions
Evaluate the six trigonometric functions of the angle 𝜃. 25 7 𝜃 24
SOLUTION
1. 𝑠𝑖𝑛= 𝑜𝑝𝑝 ℎ𝑦𝑝
2. cos= 𝑎𝑑𝑗 ℎ𝑦𝑝
3. Tan= 𝑜𝑝𝑝 𝑎𝑑𝑗
𝒔𝒊𝒏= 𝟕 𝟐𝟓
𝒄𝒐𝒔= 𝟐𝟒 𝟐𝟓
𝒕𝒂𝒏= 𝟕 𝟐𝟒
4. csc= ℎ𝑦𝑝 𝑜𝑝𝑝
5. sec= ℎ𝑦𝑝 𝑎𝑑𝑗
6. Cot= 𝑎𝑑𝑗 𝑜𝑝𝑝
𝒄𝒔𝒄= 𝟐𝟓 𝟕
𝒔𝒆𝒄= 𝟐𝟓 𝟐𝟒
𝒄𝒐𝒕= 𝟐𝟒 𝟕

6. EXAMPLE 2
Evaluate trigonometric functions
If 𝜃 is an acute angle of a right triangle and cos 𝜃= 3 8 , what is the value of csc 𝜃 ?
SOLUTION
a. 𝟑 𝟓𝟓 𝟓𝟓
b. 𝟓𝟓 𝟖
c. 𝟖 𝟓𝟓 𝟓𝟓
d. 𝟖 𝟑

7. Isosceles Right Triangle:
Geometry Conjectures
Isosceles Right Triangle:
In an isosceles right triangle, if the legs have length 𝑙, then the hypotenuse has length 𝑙 2 .
𝟑𝟎°−𝟔𝟎°−𝟗𝟎° 𝑻𝒓𝒊𝒂𝒏𝒈𝒍𝒆:
In a 30°−60°−90° triangle, if the shorter leg has length 𝑎, then the longer leg has length 𝑎 3 and the hypotenuse has length 2𝑎.

8. Vocabulary

9. EXAMPLE 3
Find an unknown side length of a right triangle
Find the value for 𝑥 in the right triangle shown. 𝑥 6
45°
SOLUTION
cos 45 𝑖𝑠 = 𝑥 6
Geometry Isosceles Conj.
𝑙 2 =6
𝑥=3 2 𝑜𝑟 4.243
𝑥=3 2 𝑜𝑟 4.243

10. 4 ∠𝑃=36° 𝑝=11.756 r=16.180 Solve ∆𝑅𝑃𝐺. 𝑅 54° 𝑔=20 𝑝 𝐺 𝑃 𝑟
EXAMPLE 4
Use a calculator to solve a right triangle
Solve ∆𝑅𝑃𝐺. 𝑅
54°
𝑔=20 𝑝 𝐺 𝑟 𝑃
SOLUTION
cos 54°= 𝑎𝑑𝑗(𝑝) 20
sin 54°= 𝑜𝑝𝑝(𝑟) 20
90°+54°+𝑃=180°
∠𝑃=36°
𝑝=11.756
r=16.180

11. EXAMPLE 4
Using indirect Measurement
Hiking 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?
𝑡𝑟𝑎𝑖𝑙 (2)
3.6 𝑚𝑖
𝑡𝑟𝑎𝑖𝑙 (3)
48°
𝑡𝑟𝑎𝑖𝑙 (1)
≈ 2.4 mi
You will travel ≈ 6.3 mi around the avalanche.

12. EXAMPLE 5
Using Angle of Elevation
KIS 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.
SOLUTION
tan 63 = 𝑥 28
𝑥≈54.953
∴The approximate height of the KIS flagpole is 55 feet.

13. Lesson 9.1 Homework:
Practice B
Practice C “Honors”