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10-3 Trigonometry
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10-3 Trigonometry
DO NOW #14
Write each fraction as a decimal rounded to the
nearest hundredth.
1.
0.67
2.
0.29
Solve each equation.
3.
4.
x = 7.25
x = 7.99
10-3 Trigonometry
Connect to Mathematical Ideas (1)(F)
By the end of today’s lesson,
SWBAT
Find the sine, cosine, and tangent of an acute
angle.
Use trigonometric ratios to find side lengths in right
triangles and to solve real-world problems.
10-3 Trigonometry
Vocabulary
trigonometric ratio
sine
cosine
tangent
10-3 Trigonometry
By the AA Similarity Postulate, a right triangle with
a given acute angle is similar to every other right triangle with
that same acute angle measure. So ∆ABC ~ ∆DEF ~ ∆XYZ,
𝐵𝐶 𝐸𝐹 𝑌𝑍
and 𝐴𝐶 = 𝐷𝐹 = 𝑋𝑍 . These are trigonometric ratios. A
trigonometric ratio is a ratio of two sides of a right triangle.
10-3 Trigonometry
Writing Math
In trigonometry, the letter of the vertex of the
angle is often used to represent the measure
of that angle. For example, the sine of A is
written as sin A.
10-3 Trigonometry
10-3 Trigonometry
Example 1: Writing Trigonometric Ratios
What are the sine, cosine, and tangent ratios for T ?
opposite
8
𝐬𝐢𝐧 𝑻 =
=
hypotenuse 17
adjacent
15
𝐜𝐨𝐬 𝑻 =
=
hypotenuse 17
opposite
𝐭𝐚𝐧 𝑻 =
adjacent
8
=
15
10-3 Trigonometry
Example 2:
Using a Trigonometric Ratios to Find Distance
Landmarks In 1990, the Leaning Tower of Pisa was
closed to the public due to safety concerns. The tower
reopened in 2001 after a 10-year project to reduce its
tilt from vertical. Engineers’ efforts were successful and
resulted in a tilt of 5o, reduced from 5.5o. Suppose
someone drops an object from the tower at a height of
150 ft. How far from the base of the tower will the object
land? Round to the nearest foot.
The given side is adjacent to the given angle. The side you
want to find its opposite the given angle.
𝑥
=
150
𝑥 = 150(𝐭𝐚𝐧 5o )
𝐭𝐚𝐧 5o
Use the tangent ratio.
Multiply each side by 150.
𝑥 ≈ 13.12329953 Use a calculator.
13 ft
10-3 Trigonometry
Caution!
Do not round until the final step of
your answer. Use the values of the
trigonometric ratios provided by your
calculator.
10-3 Trigonometry
Example 3: Using Inverses
What is the mX in the nearest degree?
You know the lengths
of the hypotenuse and
the side opposite X.
𝑼𝒔𝒆 𝒕𝒉𝒆 𝒔𝒊𝒏𝒆 𝒓𝒂𝒕𝒊𝒐.
6
𝐬𝐢𝐧 𝑿 =
10
6
−1
𝑿 = sin
10
𝒎𝑿 ≈ 37o
You know the lengths of
the hypotenuse and the
side adjacent to X.
𝑼𝒔𝒆 𝒕𝒉𝒆 𝒄𝒐𝒔𝒊𝒏𝒆 𝒓𝒂𝒕𝒊𝒐.
15
Write the ratio.
𝐜𝐨𝐬 𝑿 =
20
15
−1
𝑿 = cos
Use the inverse.
20
𝒎𝑿 ≈ 41o
Use a calculator.
10-3 Trigonometry
Example 4: Using the Tangent Ratio
An urban planner needs to know the measure
of the angle formed by a street and the edge
of a bike path. What is the measure of DEG
to the nearest degree?
∆DEG is a right triangle. You know the lengths of
the side adjacent to and the side opposite DEG.
𝑼𝒔𝒆 𝒕𝒉𝒆 𝒕𝒂𝒏𝒈𝒆𝒏𝒕 𝒓𝒂𝒕𝒊𝒐.
150
Write the ratio.
𝐭𝐚𝐧 𝐷𝐸𝐺 =
115
150
−1
DEG = tan
Use the inverse.
115
𝒎𝑫𝑬𝑮 ≈ 53o
Use a calculator.
10-3 Trigonometry
Got It ? Solve With Your Partner
Problem 1 Writing Trigonometric Ratios
What are the sine, cosine, and tangent ratios for G ?
opposite
15
𝐬𝐢𝐧 𝑮 =
=
hypotenuse 17
adjacent
8
𝐜𝐨𝐬 𝑮 =
=
hypotenuse 17
opposite
𝐭𝐚𝐧 𝑮 =
adjacent
15
=
8
10-3 Trigonometry
Got It ? Solve With Your Partner
Problem 2 Using Trigonometric Ratio to Find Distance
Find the value of w to the nearest tenth.
B.
A.
13.8
C.
1.9
3.8
10-3 Trigonometry
Got It ? Solve With Your Partner
Problem 3 Using Trigonometric Ratio to Find Distance
A section of Filbert Street in San Francisco rises
at an angle of about 17o. If you walk 150 ft up
this section, what is your vertical rise? Round to
the nearest foot.
44 ft
10-3 Trigonometry
Closure: Communicate Mathematical Ideas (1)(G)
How could you determine the value of
sin 35o without using a calculator ?
Draw a right triangle with one acute of 35o. Measure
the length of the opposite side and the hypotenuse.
Then find the ratio of the length of the opposite side to
the length of the hypotenuse.
10-3 Trigonometry
Exit Ticket: Apply Mathematics (1)(A)
Use a special right triangle to write each trigonometric ratio as a
fraction.
1. sin 60°
2. cos 45°
Use your calculator to find each trigonometric ratio. Round to the
nearest hundredth.
3. tan 84°
4. cos 13°
Find each length. Round to the nearest tenth.
5. CB
6. AC