Lesson Quiz
Lesson 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.
π=ππ π=βππ1.
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.
π .ππ²πππππππππ
1. 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 .
πππ=πππ
1.
π24
7
2.
25
πππ=ππππ
3.
πππ=πππ
πππ=πππ
4. 5.
πππ=ππππ
6.
πππ=πππ
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.
π₯=3β2ππ 4.243
cos 45 ππ β22
= π₯6
45 Β°π₯6
Geometry Isosceles Conj.
π₯=3β2ππ 4.243
Use a calculator to solve a right triangleEXAMPLE 4
SOLUTION
Solve .
π=11 .756
cos 54 Β°=ΒΏπππ (π )20
ΒΏ
54 Β°
πΊ π
π =20π
π
sin 54 Β°=ΒΏπππ (π )20
ΒΏ
r=16 .180β π=36 Β°90 Β°+54 Β°+π=180 Β°
Using indirect MeasurementEXAMPLE 4
Hiking in Nepal You are hiking toward Machapuchare βFish Tailβ 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 48Β° angle and follow that trail for 3.6 miles until you hit another trail (3) that intersects back with your original trail at a 90Β° 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?
β 2.4 mi
48 Β°π‘ππππ(1)
3.6ππ π‘ππππ(3)π‘ππππ(2)
You will travel β 6.3 mi around the avalanche.
Using Angle of ElevationEXAMPLE 5
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 63Β°. Estimate the height of the flagpole.
The approximate height of the KIS flagpole is 55 feet.
SOLUTION
π₯β54.953
tan 63=π₯28
Lesson 9.1 Homework:Practice BPractice C βHonorsβ
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