Geometry: The Tangent Ratio
This is a free lesson from our course in Geometry
In this lesson you’ll explore the definition, basic concepts of the Tangent Ratio of a right triangle and how the definition and concepts may be used to determine unknown leg lengths, to find the heights of objects like tree, flag poles, add a wheelchair ramp to the emergency exit etc. given a known side length and an angle. It also helps solve real world application problems. The presentation covering such content will be done by the instructor in own handwriting, using video and with the help of several examples and solution. This will help you understand important tangent ratio relationships to solve home work problems and also use them for real life applications. (More text below video...)
<h2> Geometry - The Tangent Ratio - Watch video </h2> <p> angle, tangent, triangle, formula, definition, geometry, opposite, example, adjacent, tangent ratio, measure, solution, right triangle, geometry help, practice questions, quizzes </p> <p> The tangent ratio is the ratio of leg opposite to an angle of a right angle to the leg adjacent to the angle</p>
People who saw this lesson also found the following lessons useful:
Inverse Tangent
Inverse Sine and Cosine
Angles of Elevation and Depression - Geometry
(Continued from above) The Tangent Ratio compares the measure of the leg opposite an angle with the measure of the leg adjacent to that angle. The symbolic representation of the tangent of angle is tan . If P is an acute angle of a right triangle and symbolically represented by ,

           = a / b
For any ratio of two numbers, if both numbers in the ratio are multiplied by the same scale factor, then the ratio is equivalent. This is also true of the tangent ratio i.e. Tangent ratio; opposite/adjacent, doesn’t change.
Thus the value of tan 53 is always the same, for triangles of any size. If you know the measures of one leg and an acute angle of a right triangle, you can use the tangent ratio to solve for the measure of the other leg.
Inverse tan
If you know the tangent ratio of an angle, finding the unknown angle is called the ‘inverse tan’ operation, and is commonly written as: tan-1. E.g. tan-1(0.6) = 30.96375653.

Look at some of the illustrative examples:
Example 1: Given- in the diagram given below, find the value of the pronumeral to two decimal places.
Step 1: tan = opposite side/adjacent side
Step 2: tan 37 = h/20
 h = 20 * tan 37 
      = 20 * 0. 7536  = 15.07 (to two decimal places)
The value is 15.07, as the final answer.
Example 2: Given- Using tangent ratio find the value of ‘a’ in triangle PQR, in the diagram below.
Step 1: mP = 20, adj leg = a in, and opposite leg = 10 in.
Step 2: tan 20 = 10/a 
            tan 20 * a = 10 
            a = 10 / tan 20
Step 3: use a calculator now: 10 = 27.47252747
Thus a = 27.5 in (rounded), as the final answer.
Example 3: Given- if tan = 5/12 and is acute, find the values of sin and cos . Draw the diagram to organize the information (Fig: below)
Step 1: tan = 5/12 (given) i.e. leg opposite to angle/leg adjacent to angle.
Step 2: hyp =(52 + 122) = 169
Step 3: Take out the sq root, it gives hyp = 13 units
Step 4: sin = opp/ hyp = 5/13, and cos = adj/hyp = 12/13. If measure of is to be worked out, draw help from trigonometric tables and find it.
 sin = 5/13, and cos = 12/13, as the final answer.
Sine and Cosine of an angle
Similarly you’ll learn to find the sine and cosine of an angle and find missing measures. In symbolic representation: Sin = a/c, and cos = b/c
Given: Measure of hypotenuse = 10 units
Measure of leg opposite = 8 units
Measure of leg adjacent = 6 units
The above lengths give: sin = 8/10= 0.8, and cos = 6/10 = 0.6.
You can find out for Q also in the similar way:
sin Q = 6/10= 0.6, and cos Q = 8/10= 0.8.
Example 1: Given- A 210 ft tall vertical tower shows a tilt by 18 ft off the perpendicular. Find the angle of tilt of the tower from vertical.
Given is: you know the length of the leg opposite the angle () i.e. 18 ft, and the length of the hypotenuse (210 ft). You can use sin A.
Step 1: To determine sin plug the known values from PQR (given).
Step 2: sin = opposite leg/hypotenuse 
                    = 18 / 210 = .0857
Step 3: In the trigonometry table, the angle for which sin is .0857, has the measure of 5 (nearest degree)
Hence angle of tilt is 5, as the final answer.
Example 2: Given- the end of an exit ramp from an interstate highway is 77 feet higher than the highway. If the ramp is 600 feet long, find value of and cos , where is the angle ramp makes with the highway.
You know the length of the leg opposite the angle () i.e. 25 ft, and the length of the hypotenuse i.e. length of the ramp (600 ft). You can use cos () formula now.
Step 1: cos () = adjacent leg/hypotenuse.
Step 2: Determine adjacent leg (a), using Pythagorean Theorem i.e.
            (600)2 = (77)2 + (a)2
Step 3: (a)2 = (600)2 - (77)2 = 354071
Step 4: Taking sq root on both the sides, a = 595 ft
Step 5: cos () = 595/600 = 0.9916.
The value of (), from trigonometric tables you can find 7
= 7 and cos () = 595/600 = 0.9916, as the final answer.
The video above will explain more in detail about The Tangent Ratio, and how to apply the concepts in solving real-world problems. This is explained with the help of several examples and done watching video.
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