Determine and Use the Sine
Ratio
Brenda Meery
Jen Kershaw
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Printed: May 23, 2016
AUTHORS
Brenda Meery
Jen Kershaw
www.ck12.org
C HAPTER
Chapter 1. Determine and Use the Sine Ratio
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Determine and Use the Sine
Ratio
In this concept, you will learn to determine and use the Sine ratio.
Jeremy was in the park flying the kite. It was a windy day and the kite was airborne within seconds. The kite was
flying at the end of a string 48 meters long, making an angle of 75◦ with the ground. How can Jeremy figure out the
altitude of his kite?
In this concept, you will learn to determine and use the Sine ratio.
Sine Ratio
You have used the TI-calculator to determine the measure of an angle using the inverse sine function (sin−1 ) when
value of the ratio was known. If sin A = 0.7845 then using the TI-calculator displayed the measure of 51.7◦ for the
angle.
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The TI-calculator can also be used to find the ratio when the measure of the angle is known. If the measure of
6 B = 48◦ then the value of the ratio can be found by following the Key Press History:
On the calculator screen the following is displayed:
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Chapter 1. Determine and Use the Sine Ratio
This decimal should always be rounded to the nearest ten thousandth (four places after the decimal).
sin 48 = 0.7431
This sine ratio for an angle of 48◦ will remain constant regardless of the size of the triangle. The sides will always
be in the same proportion to each other.
The value of the Sine ratio will always be between zero and one.
The Sine ratio is related to one of the acute angles of a right triangle such that the sine of the acute angle is the ratio
of the side opposite the specific acute angle (the reference angle) to the hypotenuse. The Sine ratio can be written as
opposite
sin 6 = hypotenuse . This equation has three parts to it - an angle and two sides. When the lengths of the two sides
are known, the measure of the angle can be calculated. When the measure of an angle and the length of one side is
known, the length of the other side can be calculated.
It is the sides of the triangle that determine the trigonometric ratio that will be used to calculate the measure of an
angle or the length of a side. To use the sine ratio, the opposite side and the hypotenuse of the right triangle must
be indicated. These two sides will have values on them if the measure of an angle is to be calculated using the sine
ratio. If the length of a side is to be calculated using the sine ratio then one of the sides will display a value and the
other will display a variable (the side that is unknown).
Let’s look at the following right triangle to see how this works.
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The measure of 6 B is 65◦ . The length of the hypotenuse is 28 meters. The side AC has the variable ’X’ on it which
means this is the side that is unknown and its length must be calculated.
The sides of the triangle can be named using the acute angle B which is the reference angle for this triangle.
The two sides that are indicated on this triangle are the hypotenuse and the opposite. The Sine ratio is the ratio of
the opposite side to the hypotenuse.
First, write the sine ratio using words.
sin B =
opposite
hypotenuse
Next, write the sine ratio using symbols.
sin B =
AC
AB
Next, fill all known values into the equation. 6 B = 65◦ ; AC = X; AB = 28.
sin 65◦ =
X
28
Next, use the TI-calculator to find the value of sin 65◦ . Round the decimal to the nearest ten thousandth.
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Chapter 1. Determine and Use the Sine Ratio
sin 65◦ = 0.9063
Next, substitute this value into the equation.
sin 65◦ =
0.9063 =
X
28
X
28
Next, multiply both sides of the equation by 28 to solve for the variable.
X
0.9063 = 28
28(0.9063) = 28
X
28
1 X
25.3764 = 28
28
25.38 = X
The answer is 25.38 meters.
The length of the opposite side of the right triangle is 25.38 meters.
When calculating the length of a side of a right triangle, the answer is usually rounded to the nearest hundredth
unless otherwise stated.
When calculating the measure of an angle of a right triangle, the answer is usually rounded to the nearest tenth unless
otherwise stated.
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Examples
Example 1
Earlier, you were given a problem about Jeremy and his kite. He needs to figure out how high his kite is above the
ground.
Jeremy can use the sine ratio to calculate the answer.
First, draw and label a right triangle to model the kite flying.
First, using the reference angle A, name the sides of the right triangle.
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Chapter 1. Determine and Use the Sine Ratio
First, write the sine ratio using words.
sin A =
opposite
hypotenuse
Next, write the sine ratio using symbols.
sin A =
BC
AB
Next, fill all known values into the equation. 6 A = 75◦ ; BC = X; AB = 48 m.
sin 75◦ =
X
48
Next, use the TI-calculator to find the value of sin 75◦ .
sin 75◦ = 0.9659
Next, substitute this value into the equation.
sin 75◦ =
0.9659 =
X
48
X
48
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Next, multiply both sides of the equation by 48 to solve for the variable.
X
0.9659 = 48
48(0.9659) = 48
X
48
1 X
46.36 = 48
48
46.36 = X
The answer is 46.36.
Jeremy’s kite is 46.36 meters above the ground.
Example 2
For the following right triangle calculate the length of side ’X’ to the nearest hundredth.
First, write down what you know from the right triangle.
F = 30◦
DE = 17.05 units
EF = X
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Next, use the reference angle F to name the sides of the triangle.
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Chapter 1. Determine and Use the Sine Ratio
The two sides that are indicated on this triangle are the hypotenuse and the opposite. The Sine ratio is the ratio of
the opposite side to the hypotenuse.
First, write the sine ratio using words.
sin F =
opposite
hypotenuse
Next, write the sine ratio using symbols.
sin F =
DE
EF
Next, fill all known values into the equation. 6 F = 30◦ ; EF = X; DE = 17.05.
sin 30◦ =
17.05
X
Next, use the TI-calculator to find the value of sin 30◦ .
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sin 30◦ = 0.5
Next, substitute this value into the equation.
sin 30◦ =
0.5 =
17.05
X
17.05
X
Next, multiply both sides of the equation by X to clear the denominator.
0.5 = 17.05
X
X(0.5) = X 17.05
X
1
X(0.5) = X 17.05
X
0.5 X = 17.05
Then, divide both sides of the equation by 0.5 to solve for the variable.
0.5 X
= 17.05
1
X
0.5
0.5
X
= 17.05
0.5
= 3.41
The answer is 34.10.
The length of the hypotenuse of the right triangle is 34.10 units.
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Chapter 1. Determine and Use the Sine Ratio
Example 3
For the following right triangle calculate the length of side ’X’.
First, write down what you know from the right triangle.
C = 32◦
BC = 15 feet
AB = X
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Next, use the reference angle C to name the sides of the triangle.
The two sides that are indicated on this triangle are the hypotenuse and the opposite. The Sine ratio is the ratio of
the opposite side to the hypotenuse.
First, write the sine ratio using words.
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sinC =
opposite
hypotenuse
Next, write the sine ratio using symbols.
sinC =
AB
BC
Next, fill all known values into the equation. 6 C = 32◦ ; AB = X; BC = 15.
sin 32◦ =
X
15
Next, use the TI-calculator to find the value of sin 32◦ .
sin 32 = 0.5299
Next, substitute this value into the equation.
sin 32◦ =
0.5299 =
X
15
X
15
Next, multiply both sides of the equation by 15 to solve for the variable.
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Chapter 1. Determine and Use the Sine Ratio
X
0.5299 = 15
15(0.5299) = 15
X
15
1 X
15(0.5299) = 15
15
7.9485 = X
7.95 = X
The answer is 7.95.
The length of the opposite side of the right triangle is 7.95 feet.
Example 4
For the following mathematical solution, using the sine ratio to determine the length of the hypotenuse of the right
triangle, briefly tell what is happening in each part of the solution.
sin B =
opposite
hypotenuse
6
sin B =
AC
BC
B = 36◦ ; AB = 11 cm; BC = X
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sin 36◦ = 11
X
0.5878 = 11
X
X(0.5878) = X 11
X
1 0.5878X = X 11
X
0.5878X = 11
0.5878
11
X = 0.5878
0.5878
X = 18.71
The answer is 18.71 cm.
The length of the hypotenuse of the right triangle is 18.71 cm.
Review
Use a calculator to find each Sine. You may round to the nearest hundredth.
1. Sine 55◦
2. Sine 25◦
3. Sine 11◦
4. Sine 60◦
5. Sine 75◦
6. Sine 12◦
7. Sine 29◦
8. Sine 15◦
Use the information given and what you have learned about trigonometric ratios to figure out the measure of each
missing side. You may round when necessary.
9. Sine angle D 2◦ , hypotenuse - 12, what is the length of the opposite side?
10. Sine angle E 65◦ , hypotenuse - 8, what is the length of the opposite side?
11. Sine angle F 45◦ , hypotenuse - 2, what is the length of the opposite side?
12. Sine angle D 25◦ , hypotenuse - 10, what is the length of the opposite side?
13. Sine angle D 80◦ , hypotenuse - 8, what is the length of the opposite side?
14. Sine angle D 45◦ , hypotenuse - 5, what is the length of the opposite side?
15. Sine angle D 40◦ , hypotenuse - 18, what is the length of the opposite side?
Answers for Review Problems
To see the Review answers, open this PDF file and look for section 7.16.
Resources
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Chapter 1. Determine and Use the Sine Ratio
MEDIA
Click image to the left or use the URL below.
URL: https://www.ck12.org/flx/render/embeddedobject/168670
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