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Chapter 7. Trigonometry
7.1 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?
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MEDIA
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Guidance
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.
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7.1. Tangent Ratio
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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
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.
Example A
Use your calculator to find the tangent of 75◦ . What does this value represent?
Solution: 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.
Example B
Solve for x.
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Chapter 7. Trigonometry
Solution: From Example A, 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
Example C
Solve for x and y.
Solution: 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
Concept Problem Revisited
As the measure of an angle increases between 0◦ and 90◦ , how does the tangent ratio of the angle change?
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7.1. Tangent Ratio
As an angle increases, the length of its opposite leg increases. Therefore,
the tangent ratio increases.
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opposite leg
ad jacent leg
increases and thus the value of
Vocabulary
Two figures are similar if a similarity transformation will carry one figure to the other. Similar figures will always
have corresponding angles congruent and corresponding sides proportional.
AA, or Angle-Angle, is a criterion for triangle similarity. The AA criterion for triangle similarity states that if two
triangles have two pairs of congruent angles, then the triangles are similar.
The tangent (tan) of an angle within a right triangle is the ratio of the length of the side opposite the angle to the
length of the side adjacent to the angle.
Guided Practice
1. 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?
2. Use your calculator to find the tangent of 45◦ . What does this value represent? Why does this value make sense?
3. Solve for x.
Answers:
1. 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.
2. 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.
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Chapter 7. Trigonometry
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.
3. 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
Practice
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.
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7.1. Tangent Ratio
6. Find the tangent of 10◦ .
7. Solve for x.
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.
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Chapter 7. Trigonometry
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◦ .
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7.2. Sine and Cosine Ratios
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7.2 Sine and Cosine Ratios
Here you will learn how to use sine and cosine ratios to find missing sides of right triangles.
SOH-CAH-TOA is a mnemonic that many people use to remember the difference between sine, cosine, and tangent.
How can remembering SOH-CAH-TOA help you?
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MEDIA
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https://www.youtube.com/watch?v=Jsiy4TxgIME Khan Academy: Basic Trigonometry
Guidance
Two right triangles with one pair of non-right congruent angles are similar by AA ∼. This means the ratio between
the side lengths of the first triangle must be congruent to the ratio between the corresponding side lengths of the
second triangle.
For example, in the picture above,
relevant ratios for a given angle.
a
c
= df . Because there are three pairs of sides for any triangle, there are three
opposite leg
1. The tangent of an angle gives the ratio adjacent leg . The abbreviation for tangent is tan.
opposite leg
2. The sine of an angle gives the ratio hypotenuse . The abbreviation for sine is sin.
adjacent leg
3. The cosine of an angle gives the ratio hypotenuse . The abbreviation for cosine is cos.
These are the basic trigonometric ratios. Trigonometry is the study of triangles. These ratios are called trigonometric ratios because they apply to triangles. Just as your scientific or graphing calculator has tangent programmed
424