Geometry One is always a long way from solving a problem until one actually has the answer. Stephen...
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Transcript of Geometry One is always a long way from solving a problem until one actually has the answer. Stephen...
Geometry
One is always a long way from solving a problem until one actually has the answer.
Stephen Hawking
Today:• 9.5
Instruction
• Practice
Geometry - Yesterday
One is always a long way from solving a problem until one actually has the answer.
Stephen Hawking
Assignment:• Trig Notes
9.5 Trigonometric Ratios
Objectives:
1. Find the sine, cosine and tangent of acute angles.
2. Use trig ratios in real life.
Vocabulary: Sine, Cosine, Tangent, Angle of Elevation
Finding Trig Ratios
A trigonometric ratio is a ratio of the lengths of two sides of a right triangle. The word trigonometry is derived from the ancient Greek language and means measurement of triangles. The three basic trigonometric ratios are sine, cosine, and tangent, which are abbreviated as sin, cos, and tan respectively.
Trigonometric Ratios
Let ∆ABC be a right triangle. The since, the cosine, and the tangent of the acute angle A are defined as follows.
ac
bside adjacent to angle A
Sideoppositeangle A
hypotenuse
A
B
C
sin A =Side opposite A
hypotenuse= a
c
cos A =Side adjacent A
hypotenuse= b
c
tan A =Side opposite A
Side adjacent A
= a
b
Ex. 1: Finding Trig Ratios
Compare the sine, the cosine, and the tangent ratios for A in each triangle beside.
By the SSS Similarity Theorem, the triangles are similar. Their corresponding sides are in proportion which implies that the trigonometric ratios for A in each triangle are the same.
15
817
A
B
C
7.5
48.5
A
B
C
Ex. 1: Finding Trig Ratios
15
817
A
B
C
7.5
48.5
A
B
C
Large Small
sin A = opposite
hypotenuse
cosA = adjacent
hypotenuse
tanA = opposite
adjacent
8
17≈ 0.4706
15
17≈ 0.8824
8
15≈ 0.5333
4
8.5≈ 0.4706
7.5
8.5≈ 0.8824
4
7.5≈ 0.5333
Trig ratios can be expressed as fractions or decimals.
Finding Trig Ratios - On Your OWN
15
817
A
B
C
sin B = opposite
hypotenuse
cosB = adjacent
hypotenuse
tanB = opposite
adjacent
Take the big triangle from example #1, now focus on angle B.
How does this change your labels?
Using a Calculator
You can use a calculator to approximate the sine, cosine, and the tangent of 74. Make sure that your calculator is in degree mode. The table shows some sample keystroke sequences accepted by most calculators.
Sample keystrokes
Sample keystroke sequences
Sample calculator display Rounded
Approximation
74
74
0.961262695 0.9613
0.275637355 0.2756
3.487414444 3.4874
sinsin
ENTER
74
74
COS
COS
ENTER
74
74
TAN
TANENTER
Using Trigonometric Ratios in Real-life
Suppose you stand and look up at a point in the distance. Maybe you are looking up at the top of a tree as in Example 6. The angle that your line of sight makes with a line drawn horizontally is called angle of elevation.
Indirect Measurement You are measuring the
height of a Sitka spruce tree in Alaska. You stand 45 feet from the base of the tree. You measure the angle of elevation from a point on the ground to the top of the top of the tree to be 59°. To estimate the height of the tree, you can write a trigonometric ratio that involves the height h and the known length of 45 feet.
The math
tan 59° =opposite
adjacent
tan 59° =h
4545 tan 59° = h
45 (1.6643) ≈ h
74.9 ≈ h
Write the ratio
Substitute values
Multiply each side by 45
Use a calculator or table to find tan 59°
Simplify
The tree is about 75 feet tall.
Estimating Distance
Escalators. The escalator at the Wilshire/Vermont Metro Rail Station in Los Angeles rises 76 feet at a 30° angle. To find the distance d a person travels on the escalator stairs, you can write a trigonometric ratio that involves the hypotenuse and the known leg of 76 feet.
d76 ft
30°
Now the math
d76 ft
30°
sin 30° =oppositehypotenuse
sin 30° =76
d
d sin 30° = 76
sin 30° 76
d =
0.5 76
d =
d = 152
Write the ratio for sine of 30°
Substitute values.
Multiply each side by d.
Divide each side by sin 30°
Substitute 0.5 for sin 30°
Simplify
A person travels 152 feet on the escalator stairs.