Writing Trigonometric Ratios
Warm-Up
A trigonometric ratio is a ratio of two sides of a right triangle. We can use
trigonometric ratios to relate the angles of a triangle to the lengths of the triangle’s sides.
In a right triangle, three relationships exist which relate the angles of a triangle to the
measures of its sides.
The word opposite means _________________________________________________
The word adjacent means _________________________________________________
The hypotenuse is _______________________________________________________
sine (sin) of Angle x =
𝑆𝑖𝑑𝑒 𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑡𝑜 𝐴𝑛𝑔𝑙𝑒 𝑥
cosine (cos) of Angle x =
𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑆𝑖𝑑𝑒 𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑡𝑜 𝐴𝑛𝑔𝑙𝑒 𝑥
tangent (tan) of Angle x =
𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑆𝑖𝑑𝑒 𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑡𝑜 𝐴𝑛𝑔𝑙𝑒 𝑥
𝑆𝑖𝑑𝑒 𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑡𝑜 𝐴𝑛𝑔𝑙𝑒 𝑥
We write:
sin 𝑥 =
𝑜𝑝𝑝 𝑥
ℎ𝑦𝑝
cos 𝑥 =
𝑎𝑑𝑗 𝑥
ℎ𝑦𝑝
tan 𝑥 =
𝑜𝑝𝑝 𝑥
𝑎𝑑𝑗 𝑥
Remember this!---------------------------------------->
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Remember to label all sides first:
OPPOSITE, ADJACENT,
HYPOTENUSE
Example
For the given triangle, express as a fraction:
a) sin A
b) cos A
c) tan A
For the given triangle, express as a fraction:
d) sin B
e) cos B
f) tan B
Express each of the above as a decimal rounded to the nearest ten-thousandth:
sin B = _______________ cos B = ___________________ tan B = _______________
Sides OPPOSITE and
ADJACENT will change
depending on which angle
you’re talking about.
The HYPOTENUSE will
always be across from the
right angle. So label it
first!
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Exercise
1) For the given triangle, express as a fraction:
a) sin K
b) cos K
c) tan K
d) sin J
e) cos J
f) tan J
2) For the given triangle, express as a fraction and a decimal rounded to the nearest
ten-thousandth:
a) cos L
b) tan L
c) sin L
d) sin M
e) tan M
f) cos M
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Finding Values on the Calculator
The sine, cosine, and tangent of any angle is a known value that can be found using any
scientific calculator.
Examples
Using your calculator, find the value of each to the nearest ten-thousandth:
𝑠𝑖𝑛 38° _______________ 𝑐𝑜𝑠 38° _______________tan 38° ___________________
Exercise
Using your calculator, find the value of each to the nearest ten-thousandth:
𝑡𝑎𝑛 22° _______________ 𝑐𝑜𝑠 17° _______________tan 31° ___________________
𝑠𝑖𝑛 80° _______________ 𝑠𝑖𝑛 42° _______________cos 60° ___________________
𝑠𝑖𝑛 45° _______________𝑡𝑎𝑛 65° _______________cos 44° ___________________
Lesson Summary
The Three Trigonometric Ratios
sin 𝑥 =
𝑜𝑝𝑝 𝑥
ℎ𝑦𝑝
cos 𝑥 =
𝑎𝑑𝑗 𝑥
ℎ𝑦𝑝
tan 𝑥 =
𝑜𝑝𝑝 𝑥
𝑎𝑑𝑗 𝑥
Exit ticket
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Homework
Express each ratio indicated as a fraction.
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Finding a Missing Side Length Using the Trigonometric Ratios
Warm-Up
The trigonometric ratios can help us to solve for a missing side in a right triangle.
sin 𝑥 =
𝑜𝑝𝑝 𝑥
ℎ𝑦𝑝
cos 𝑥 =
𝑎𝑑𝑗 𝑥
ℎ𝑦𝑝
tan 𝑥 =
𝑜𝑝𝑝 𝑥
𝑎𝑑𝑗 𝑥
Example #1
Find, to the nearest tenth:
a) AB
b) AC
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Example #2
Find, to the nearest tenth:
a) MP
b) NM
Exercise
1) Find, to the nearest tenth:
a) XY
b) YZ
2) Find, to the nearest tenth:
a) DE
b) DF
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3) Find, to the nearest tenth:
a) TU
b) TV
Solving Problems Using Trigonometric Ratios
Model Problem
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Exercise
3) Find KL.
4) Find GH and HJ.
Lesson Summary
1. Label the sides of the triangle, OPPOSITE, ADJACENT, and HYPOTENUSE.
2. Using SOH-CAH-TOA, determine which trig ratio to use in the problem.
3. Set up the trig ratio using the formula.
4. Cross-multiply and solve.
Exit Ticket
Find JL.
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Homework
7) Find TV and TU.
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2)
3)
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Finding a Missing Angle When the Side Lengths are Known
Warm-Up
Finding Missing Angle Measures Using Inverse Functions
If you know the sine, cosine, or tangent of an acute angle measure, you can use the
inverse trigonometric functions to find the measure of the angle.
We say
“sine
inverse”
Example #1
Find sin 35° ___________________________ Find sin−1(.573576) _________________
Find cos 18° ___________________________ Find cos −1(.951056) _________________
Find tan 22° ___________________________ Find sin−1(.404026) _________________
What does the inverse function do? _____________________________________
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Example #2
Find the measure of angle A if:
1) sin A = 0.3456
2) tan A = 1.4552
3) cos A = 0.4995
Example #3
Find the measure of angle A to the nearest degree.
Guided Practice
Find the measure of angle P to the nearest degree.
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Practice
Find each angle measure to the nearest degree.
1) Find m∠𝑅.
3) Find m∠𝐹.
2) Find m∠𝐵.
4) Find m∠𝑅 .
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Solving Word Problems
Model Problem
Exercise
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Homework
13)
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