Tangent Ratio
CK-12
Kaitlyn Spong
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Printed: April 29, 2016
AUTHORS
CK-12
Kaitlyn Spong
www.ck12.org
C HAPTER
Chapter 1. Tangent Ratio
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Tangent Ratio
Here you will learn how to use the tangent ratio to find missing sides of right triangles.
As the measure of an angle increases between 0◦ and 90◦ , how does the tangent ratio of the angle change?
Tangent Ratio
Recall that one way to show that two triangles are similar is to show that they have two pairs of congruent angles.
This means that two right triangles will be similar if they have one pair of congruent non-right angles.
The two right triangles above are similar because they have two pairs of congruent angles. This means that their
corresponding sides are proportional. DF and AC are corresponding sides because they are both opposite the 22◦
4
FE
10
angle. DF
AC = 2 = 2, so the scale factor between the two triangles is 2. This means that x = 10, because CB = 5 = 2.
The ratio between the two legs of any 22◦ right triangle will always be the same, because all 22◦ right triangles are
similar. The ratio of the length of the leg opposite the 22◦ angle to the length of the leg adjacent to the 22◦ angle
will be 52 = 0.4. You can use this fact to find a missing side of another 22◦ right triangle.
Because this is a 22◦ right triangle, you know that
opposite leg
ad jacent leg
=
2
5
= 0.4.
opposite leg
= 0.4
ad jacent leg
7
= 0.4
x
0.4x = 7
x = 17.5
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The ratio between the opposite leg and the adjacent leg for a given angle in a right triangle is called the tangent ratio.
opposite leg
Your scientific or graphing calculator has tangent programmed into it, so that you can determine the ad
jacent leg ratio
for any angle within a right triangle. The abbreviation for tangent is tan.
MEDIA
Click image to the left or use the URL below.
URL: https://www.ck12.org/flx/render/embeddedobject/75767
Solve the following problem
Use your calculator to find the tangent of 75◦ . What does this value represent?
Make sure your calculator is in degree mode. Then, type “tan(75)”.
tan(75◦ ) ≈ 3.732
This means that the ratio of the length of the opposite leg to the length of the adjacent leg for a 75◦ angle within a
right triangle will be approximately 3.732.
Solve for
From the previous problem, you know that the ratio
opposite leg
ad jacent leg
≈ 3.732. You can use this to solve for x.
opposite leg
≈ 3.732
ad jacent leg
x
≈ 3.732
2
x ≈ 7.464
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Chapter 1. Tangent Ratio
Solve for
You can use the 65◦ angle to find the correct ratio between 24 and x.
opposite leg
ad jacent leg
24
2.145 ≈
x
24
x≈
2.145
x ≈ 11.189
tan(65◦ ) =
Note that this answer is only approximate because you rounded the value of tan 65◦ . An exact answer will include
“tan”. The exact answer is:
x=
24
tan 65◦
To solve for y, you can use the Pythagorean Theorem because this is a right triangle.
11.1892 + 242 = y2
701.194 = y2
26.48 = y
Examples
Example 1
Earlier, you were asked how does the tangent ratio of the angle change.
As the measure of an angle increases between 0◦ and 90◦ , how does the tangent ratio of the angle change?
As an angle increases, the length of its opposite leg increases. Therefore,
the tangent ratio increases.
opposite leg
ad jacent leg
increases and thus the value of
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Example 2
Tangent tells you the ratio of the two legs of a right triangle with a given angle. Why does the tangent ratio not work
in the same way for non-right triangles?
Two right triangles with a 32◦ angle will be similar. Two non-right triangles with a 32◦ angle will not necessarily
be similar. The tangent ratio works for right triangles because all right triangles with a given angle are similar. The
tangent ratio doesn’t work in the same way for non-right triangles because not all non-right triangles with a given
angle are similar. You can only use the tangent ratio for right triangles.
Example 3
Use your calculator to find the tangent of 45◦ . What does this value represent? Why does this value make sense?
tan(45◦ ) = 1. This means that the ratio of the length of the opposite leg to the length of the adjacent leg is equal to
1 for right triangles with a 45◦ angle.
This should make sense because right triangles with a 45◦ angle are isosceles. The legs of an isosceles triangle are
congruent, so the ratio between them will be 1.
Example 4
Solve for x.
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Chapter 1. Tangent Ratio
Use the tangent ratio of a 35◦ angle.
opposite leg
ad jacent leg
x
tan(35◦ ) =
18
x = 18 tan(35◦ )
tan(35◦ ) =
x ≈ 12.604
Review
1. Why are all right triangles with a 40◦ angle similar? What does this have to do with the tangent ratio?
2. Find the tangent of 40◦ .
3. Solve for x.
4. Find the tangent of 80◦ .
5. Solve for x.
6. Find the tangent of 10◦ .
7. Solve for x.
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8. Your answer to #5 should be the same as your answer to #7. Why?
9. Find the tangent of 27◦ .
10. Solve for x.
11. Find the tangent of 42◦ .
12. Solve for x.
13. A right triangle has a 42◦ angle. The base of the triangle, adjacent to the 42◦ angle, is 5 inches. Find the area of
the triangle.
√
14. Recall that the ratios between the sides of a 30-60-90 triangle are 1 : 3 : 2. Find the tangent of 30◦ . Explain
how this matches the ratios for a 30-60-90 triangle.
15. Explain why it makes sense that the value of the tangent ratio increases as the angle goes from 0◦ to 90◦ .
Answers for Review Problems
To see the Review answers, open this PDF file and look for section 7.1.
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Chapter 1. Tangent Ratio
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