Transcript Right Angle - Biloxi Public Schools

10-1 Right-Angle Trigonometry
Given the measure of one of the acute
angles in a right triangle, find the
measure of the other acute angle.
1. 45° 45°
2. 60° 30°
3. 24° 66°
4. 38°
Holt McDougal Algebra 2
52°
10-1 Right-Angle Trigonometry
Application Lab
• Pythagorean Theorem activity 1
Holt McDougal Algebra 2
10-1 Right-Angle Trigonometry
Find the unknown length for each right
triangle with legs a and b and hypotenuse
c.
5. b = 12, c =13
6. a = 3, b = 3
Holt McDougal Algebra 2
a=5
10-1 Right-Angle Trigonometry
Holt McDougal Algebra 2
10-1 Right-Angle Trigonometry
Caution!
Make sure that your graphing calculator is set to
interpret angle values as degrees. Press
.
Check that Degree and not Radian is
highlighted in the third row.
Holt McDougal Algebra 2
10-1 Right-Angle Trigonometry
Objectives
Understand and use trigonometric
relationships of acute angles in
triangles.
Determine side lengths of right
triangles by using trigonometric
functions.
Holt McDougal Algebra 2
10-1 Right-Angle Trigonometry
Vocabulary
trigonometric function
sine
cosine
tangent
Holt McDougal Algebra 2
10-1 Right-Angle Trigonometry
A trigonometric function is a function whose
rule is given by a trigonometric ratio.
A trigonometric ratio compares the lengths of two
sides of a right triangle.
The Greek letter theta θ is traditionally used to
represent the measure of an acute angle in a
right triangle.
The values of trigonometric ratios depend upon
θ.
Holt McDougal Algebra 2
10-1 Right-Angle Trigonometry
Holt McDougal Algebra 2
10-1 Right-Angle Trigonometry
Example 1: Finding Trigonometric Ratios
Find the value of the
sine, cosine, and
tangent functions for θ.
sin θ =
cos θ =
tan θ =
Holt McDougal Algebra 2
10-1 Right-Angle Trigonometry
Check It Out! Example 2
Find the value of the
sine, cosine, and
tangent functions for θ.
sin θ =
cos θ =
tan θ =
Holt McDougal Algebra 2
10-1 Right-Angle Trigonometry
Application Lab
• Pythagorean Theorem activity: Walk
though the Desert:, change to sine,
cosine, tangent
Holt McDougal Algebra 2
10-1 Right-Angle Trigonometry
Holt McDougal Algebra 2
10-1 Right-Angle Trigonometry
You will frequently need to determine the value of
trigonometric ratios for 30°,60°, and 45° angles as
you solve trigonometry problems. Recall from
geometry that in a 30°-60°-90° triangle, the
ration of the side lengths is 1: 3 :2, and that in a
45°-45°-90° triangle, the ratio of the side lengths is
1:1: 2.
Holt McDougal Algebra 2
10-1 Right-Angle Trigonometry
Holt McDougal Algebra 2
10-1 Right-Angle Trigonometry
Example 2: Finding Side Lengths of Special Right
Triangles
Use a trigonometric function to find the value of x.
°
The sine function
relates the opposite
leg and the
hypotenuse.
Substitute 30° for θ, x for
opp, and 74 for hyp.
Substitute
x = 37
Holt McDougal Algebra 2
for sin 30°.
Multiply both sides by 74 to solve for x.
10-1 Right-Angle Trigonometry
Check It Out! Example 2
Use a trigonometric function to find the value of x.
°
The sine function
relates the opposite
leg and the
hypotenuse.
Substitute 45 ° for θ, x for
opp, and 20 for hyp.
Substitute
for sin 45°.
Multiply both sides by 20 to solve for x.
Holt McDougal Algebra 2
10-1 Right-Angle Trigonometry
Example 3: Sports Application
In a waterskiing competition,
a jump ramp has the
measurements shown. To
the nearest foot, what is
the height h above water
that a skier leaves the ramp?
Substitute 15.1° for θ, h for opp., and 19
for hyp.
5≈h
Multiply both sides by 19.
Use a calculator to simplify.
The height above the water is about 5 ft.
Holt McDougal Algebra 2
10-1 Right-Angle Trigonometry
Check It Out! Example 3
A skateboard ramp will
have a height of 12 in.,
and the angle between
the ramp and the ground
will be 17°. To the nearest inch, what will
be the length l of the ramp?
Substitute 17° for θ, l for hyp., and 12
for opp.
Multiply both sides by l and divide by
sin 17°.
Use a calculator to simplify.
l ≈ 41
The length of the ramp is about 41 in.
Holt McDougal Algebra 2
10-1 Right-Angle Trigonometry
Example 4: Geology Application
A biologist whose eye level is 6 ft above the
ground measures the angle of elevation to the
top of a tree to be 38.7°. If the biologist is
standing 180 ft from the tree’s base, what is
the height of the tree to the nearest foot?
Step 1 Draw and label a
diagram to represent the
information given in the
problem.
Holt McDougal Algebra 2
10-1 Right-Angle Trigonometry
Example 4 Continued
Step 2 Let x represent the height of the tree
compared with the biologist’s eye level.
Determine the value of x.
Use the tangent function.
Substitute 38.7 for θ, x for opp., and
180 for adj.
180(tan 38.7°) = x
Multiply both sides by 180.
144 ≈ x Use a calculator to solve for x.
Holt McDougal Algebra 2
10-1 Right-Angle Trigonometry
Example 4 Continued
Step 3 Determine the overall height of the
tree.
x + 6 = 144 + 6
= 150
The height of the tree is about 150 ft.
Holt McDougal Algebra 2
10-1 Right-Angle Trigonometry
Check It Out! Example 4
A surveyor whose eye level is 6 ft above the
ground measures the angle of elevation to the
top of the highest hill on a roller coaster to be
60.7°. If the surveyor is standing 120 ft from
the hill’s base, what is the height of the hill to
the nearest foot?
Step 1 Draw and label a
diagram to represent the
information given in the
problem.
Holt McDougal Algebra 2
60.7°
120 ft
10-1 Right-Angle Trigonometry
Check It Out! Example 4 Continued
Step 2 Let x represent the height of the hill
compared with the surveyor’s eye level.
Determine the value of x.
Use the tangent function.
Substitute 60.7 for θ, x for opp., and
120 for adj.
120(tan 60.7°) = x
Multiply both sides by 120.
214 ≈ x Use a calculator to solve for x.
Holt McDougal Algebra 2
10-1 Right-Angle Trigonometry
Check It Out! Example 4 Continued
Step 3 Determine the overall height of the
roller coaster hill.
x + 6 = 214 + 6
= 220
The height of the hill is about 220 ft.
Holt McDougal Algebra 2
10-1 Right-Angle Trigonometry
Holt McDougal Algebra 2
10-1 Right-Angle Trigonometry
trigonometric function
• sine
• cosine
• tangent
• cosecants
• secant
• cotangent
Vocabulary
Holt McDougal Algebra 2
10-1 Right-Angle Trigonometry
Helpful Hint
In each reciprocal pair of trigonometric functions,
there is exactly one “co”
Holt McDougal Algebra 2
10-1 Right-Angle Trigonometry
The reciprocals of the sine, cosine, and tangent
ratios are also trigonometric ratios. They are
trigonometric functions, cosecant, secant, and
cotangent.
Holt McDougal Algebra 2
10-1 Right-Angle Trigonometry
Example 5: Finding All Trigonometric Functions
Find the values of the six trigonometric
functions for θ.
Step 1 Find the length of the hypotenuse.
a2 + b2 = c2
c2 = 242 + 702
Pythagorean Theorem.
Substitute 24 for a and
70 for b.
c2 = 5476
Simplify.
c = 74
Holt McDougal Algebra 2
Solve for c. Eliminate
the negative
solution.
70
θ
24
10-1 Right-Angle Trigonometry
Example 5 Continued
Step 2 Find the function values.
Holt McDougal Algebra 2
10-1 Right-Angle Trigonometry
Check It Out! Example 5
Find the values of the six trigonometric
functions for θ.
Step 1 Find the length of the hypotenuse.
a2 + b2 = c2
c2 = 182 + 802
Pythagorean Theorem.
Substitute 18 for a and
80 for b.
c2 = 6724
Simplify.
c = 82
Holt McDougal Algebra 2
Solve for c. Eliminate
the negative
solution.
80
θ
18
10-1 Right-Angle Trigonometry
Check It Out! Example 5 Continued
Step 2 Find the function values.
Holt McDougal Algebra 2
10-1 Right-Angle Trigonometry
Lesson Quiz: Part I
Solve each equation. Check your answer.
1. Find the values of the six trigonometric functions
for θ.
Holt McDougal Algebra 2
10-1 Right-Angle Trigonometry
Lesson Quiz: Part II
2. Use a trigonometric function to find the value
of x.
3. A helicopter’s altitude is 4500 ft, and a plane’s
altitude is 12,000 ft. If the angle of depression
from the plane to the helicopter is 27.6°, what is
the distance between the two, to the nearest
hundred feet?
16,200 ft
Holt McDougal Algebra 2