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Tangent Identification

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Tangent Ratio
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As the measure of an angle increases between 0^\circ  and 90^\circ , how does the tangent ratio of the angle change?

Watch This

https://www.youtube.com/watch?v=cquWNuKTnQs Trigonometric Ratios: Tangent

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.  \overline{DF} and \overline{AC}  are corresponding sides because they are both opposite the  22^\circ angle. \frac{DF}{AC}=\frac{4}{2}=2 , so the scale factor between the two triangles is 2. This means that x=10 , because \frac{FE}{CB}=\frac{10}{5}=2 .

The ratio between the two legs of any 22^\circ  right triangle will always be the same, because all 22^\circ  right triangles are similar. The ratio of the length of the leg opposite the 22^\circ  angle to the length of the leg adjacent to the 22^\circ  angle will be \frac{2}{5}=0.4 . You can use this fact to find a missing side of another 22^\circ  right triangle.

Because this is a 22^\circ  right triangle, you know that \frac{opposite \ leg}{adjacent \ leg}=\frac{2}{5}=0.4 .

\frac{opposite \ leg}{adjacent \ leg} &= 0.4\\\frac{7}{x} &= 0.4\\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 . Your scientific or graphing calculator has tangent programmed into it, so that you can determine the \frac{opposite \ leg}{adjacent \ 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^\circ . What does this value represent?

Solution: Make sure your calculator is in degree mode. Then, type “ \tan (75) ”.

\tan (75^\circ) \approx 3.732

This means that the ratio of the length of the opposite leg to the length of the adjacent leg for a 75^\circ  angle within a right triangle will be approximately 3.732.

Example B

Solve for x .

Solution: From Example A, you know that the ratio \frac{opposite \ leg}{adjacent \ leg} \approx 3.732 . You can use this to solve for x .

\frac{opposite \ leg}{adjacent \ leg} & \approx 3.732\\\frac{x}{2} & \approx 3.732\\x & \approx 7.464

Example C

Solve for x  and y .

Solution:  You can use the 65^\circ  angle to find the correct ratio between 24 and x .

\tan (65^\circ) &= \frac{opposite \ leg}{adjacent \ leg}\\2.145 & \approx \frac{24}{x}\\x & \approx \frac{24}{2.145}\\x & \approx 11.189

Note that this answer is only approximate because you rounded the value of \tan 65^\circ . An exact answer will include “ \tan ”. The exact answer is:

x=\frac{24}{\tan 65^\circ}

To solve for y , you can use the Pythagorean Theorem because this is a right triangle.

11.189^2+24^2 &= y^2\\701.194 &= y^2\\26.48 &= y

Concept Problem Revisited

As the measure of an angle increases between 0^\circ  and 90^\circ , how does the tangent ratio of the angle change?

As an angle increases, the length of its opposite leg increases. Therefore, \frac{opposite \ leg}{adjacent \ leg}  increases and thus the value of the tangent ratio increases. 

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^\circ . What does this value represent? Why does this value make sense?

3. Solve for x .

Answers:

1. Two right triangles with a 32^\circ  angle will be similar. Two non-right triangles with a 32^\circ  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^\circ)=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^\circ  angle.

This should make sense because right triangles with a 45^\circ  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^\circ  angle.

\tan (35^\circ) &= \frac{opposite \ leg}{adjacent \ leg}\\\tan (35^\circ) &= \frac{x}{18}\\x &= 18 \tan (35^\circ)\\x & \approx 12.604

Practice

1. Why are all right triangles with a 40^\circ  angle similar? What does this have to do with the tangent ratio?

2. Find the tangent of 40^\circ .

3. Solve for x .

4. Find the tangent of 80^\circ .

5. Solve for x .

6. Find the tangent of 10^\circ .

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^\circ .

10. Solve for x .

11. Find the tangent of 42^\circ .

12. Solve for x .

13. A right triangle has a 42^\circ  angle. The base of the triangle, adjacent to the 42^\circ  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:\sqrt{3} : 2 . Find the tangent of 30^\circ . 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^\circ  to 90^\circ .

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