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8.4 Trigonometric Ratios- Sine and Cosine Geometry Mr. Peebles Spring 2013
8.4 Trigonometric
Ratios- Sine and
Cosine Geometry
Mr. Peebles
Spring 2013
Daily Learning Target (DLT)
Wednesday April 17, 2013 • “I can apply my knowledge of right triangles
to find the sine and cosine of an acute angle.”
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
Side
opposite
angle A
hypotenuse
A
B
C
sin A = Side opposite A
hypotenuse
= a
c
cos A = Side adjacent to A
hypotenuse
= b
c
tan A = Side opposite A
Side adjacent to A
= a
b
Trigonometric Ratios
• TOA Tangent = Opposite/Adjacent
• CAH Cosine = Adjacent/Hypotenuse
• SOH Sine= Opposite/Hypotenuse
Trigonometric Ratios
• TOA “Together Only Actors
• SOH Sing On Holidays
• CAH Cheering All Happily.”
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
Trig ratios are often expressed as decimal approximations.
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
8
17 ≈ 0.4706
15
17 ≈ 0.8824
4
8.5 ≈ 0.4706
7.5
8.5 ≈ 0.8824
Trig ratios are often expressed as decimal approximations.
Ex. 2: Finding Trig Ratios
S
sin S = opposite
hypotenuse
cosS = adjacent
hypotenuse
adjacent
opposite
12
13 hypotenuse5
R
T S
Ex. 2: Finding Trig Ratios
S
sin S = opposite
hypotenuse
cosS = adjacent
hypotenuse
5
13 ≈ 0.3846
12
13 ≈ 0.9231
adjacent
opposite
12
13 hypotenuse5
R
T S
Ex. 2: Finding Trig Ratios—Find the sine,
the cosine, and the tangent of the
indicated angle.
R
sin S = opposite
hypotenuse
cosS = adjacent
hypotenuse
adjacent
opposite12
13 hypotenuse5
R
T S
Ex. 2: Finding Trig Ratios—Find the sine,
the cosine, and the tangent of the
indicated angle.
R
sin S = opposite
hypotenuse
cosS = adjacent
hypotenuse
12
13 ≈ 0.9231
5
13 ≈ 0.3846
adjacent
opposite12
13 hypotenuse5
R
T S
Ex. 3: Finding Trig Ratios—Find the sine,
the cosine, and the tangent of 45
45
sin 45= opposite
hypotenuse
adjacent
hypotenuse
1
hypotenuse1
√2
cos 45=
Begin by sketching a 45-45-90 triangle. Because all such triangles are similar, you can make calculations simple by choosing 1 as the length of each leg. From Theorem 9.8 on page 551, it follows that the length of the hypotenuse is √2.
45
Ex. 3: Finding Trig Ratios—Find the sine,
the cosine, and the tangent of 45
45
sin 45= opposite
hypotenuse
adjacent
hypotenuse
1
hypotenuse1
√2
cos 45= 1
√2 =
√2
2 ≈ 0.7071
1
√2 =
√2
2 ≈ 0.7071
Begin by sketching a 45-45-90 triangle. Because all such triangles are similar, you can make calculations simple by choosing 1 as the length of each leg. From Theorem 9.8 on page 551, it follows that the length of the hypotenuse is √2. 45
2
1
Ex. 4: Finding Trig Ratios—Find the sine,
the cosine, and the tangent of 30
30
sin 30= opposite
hypotenuse
adjacent
hypotenuse
√3
cos 30=
Begin by sketching a 30-60-90 triangle. To make the calculations simple, you can choose 1 as the length of the shorter leg. From Theorem 9.9, on page 551, it follows that the length of the longer leg is √3 and the length of the hypotenuse is 2. 30
2
1
Ex. 4: Finding Trig Ratios—Find the sine,
the cosine, and the tangent of 30
30
sin 30= opposite
hypotenuse
adjacent
hypotenuse
√3
cos 30= √3
2 ≈ 0.8660
1
2 = 0.5
Begin by sketching a 30-60-90 triangle. To make the calculations simple, you can choose 1 as the length of the shorter leg. From Theorem 9.9, on page 551, it follows that the length of the longer leg is √3 and the length of the hypotenuse is 2. 30
Sample keystrokes Sample keystroke
sequences
Sample calculator display Rounded
Approximation
0.275637355 0.2756 sin
sin
ENTER
COS
Sample keystrokes Sample keystroke
sequences
Sample calculator display Rounded
Approximation
3.487414444 3.4874 COS
ENTER
TAN
Notes:
• If you look back at Examples 1-4, you will notice that the sine or the cosine of an acute triangles is always less than 1. The reason is that these trigonometric ratios involve the ratio of a leg of a right triangle to the hypotenuse. The length of a leg or a right triangle is always less than the length of its hypotenuse, so the ratio of these lengths is always less than one.
Ex. 5: 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° =
opposite
hypotenuse
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.
Assignment:
• 1. Pgs. 434-436 (2-20 Evens, 31-34, 35-43 Odds).
• 2. Complete 30-60-90 Triangle Worksheet
• 3. Complete Rationalizing Denominators
• 4. Pgs. 441-442 (1-17 Odds, 22-24)
Exit Quiz – 10 Points
20°
x
6 cm
y
Solve for x and y. PLEASE
SHOW ALL WORK.
