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## Get ready for Algebra 2

### Course: Get ready for Algebra 2>Unit 5

Lesson 5: Solving for sides and angles in right triangles using the trigonometric ratios

# Solving for a side in right triangles with trigonometry

Sal is given a right triangle with an acute angle of 65° and a leg of 5 units, and he uses trigonometry to find the two missing sides. Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

• At he says " you could of solved this using the Pythagorean theorem... But there is an issue if im not mistaken:

10.7*10.7 + 5*5 does not equal to 11.8? •   I don't think there is an issue he just wanted to solve using the trigonometric functions because that is what we had just been working on.

Just to show, 10.7*10.7 = 114.49
5*5 = 25
114.49+25=139.49
And the square root of 139.49 = 11.8

a^2+b^2=c^2 so don't forget to square root everything.
• How would you find tan65 degrees without a calculator? •   You will not be asked to do that computation: It is exceedingly difficult and requires knowing advanced mathematics. Thus, the actual computations are not taught at this level of study. If you are not allowed a calculator on a test or something, you should be given the needed values.

If you are given the sin and cosine of an angle, then the tangent is just sin x/ cos x.

Anyway, just for reference, here is the actual computation for tan 65°:
-i [e^(-25iπ/180) + e^(25iπ/180)] ÷ [ e^(-25iπ/180)-e^(25iπ/180)]
Obviously, doing the calculation by hand is going to be too difficult.

You will, however, be expected to memorize the sin, cos, and tan of certain special angles such as 0, 30°, 45°, 60°, 90°, etc.
• Are the tangent, sine, and cosine "fixed" ratios depending on the angle? Is that why you can just put in tan 65 into a calculator and have it give you a number? • I'm having trouble understanding the solution to a problem in the section on Trigonometry 2 questions. I took a screenshot of the problem with all its hints revealed: http://s14.postimg.org/gq06og00h/Screenshot_2014_01_11_at_11_24_57_AM.png.

The issue I'm having is that I can't seem to wrap my head around the last part of the hints. How does one get from [10 / (10 times the square root of 109) / 109] to an answer of "square root of 109"? I must be missing something algebraically, or in my order of operations. I've been stuck on this for little while now, so I'd appreciate any clarification you can offer. Thanks! • In my calculator, when I do 5/cos(65). I am getting approx. -8.8896, and Sal got approximately 11.8. Can somebody tell me what I am doing wrong? Do you need to do something with calculator for this to work? •  This is a common issue with calculators, yours currently is set in radian mode, so you will have to change it to degree mode to get correct answers when working in degrees. TI calculators have a mode button to change it.
• Is there a way to find the sides is if you have just angles given, or the measure of the angles if just the sides are given? • If you only have the angles, there is no way to find the sides. There is an infinite number of triangles fitting any three angles that add up to 180 degrees.

If you know all the sides, you can figure out everything about that triangle.
• how can you derive values for cos and sin and other trig ratios without using a calculator? after all when no calculators were there trig was invented and solved to utmost presicion? • It wasn't really utmost precision through much of history, but you're right that it was remarkably good. We credit that to Ptolemy, who devised his trig tables in the second century by half-degrees that turned out to be correct to the fifth or sixth decimal place for the most part. His exact path is hard to reproduce and would involve geometry that is more detailed than a typical high-school education, but a modern person following in Ptolemy's footsteps would get the same results from using the sum-angle and half-angle identities that you will learn shortly plus the known values of several critical values like 60 degrees and 72 degrees.
• how to calculate these values without using a calculator? • Which values? If you mean the values of the trigonometric functions, then that math is too difficult to do by hand. The reason the actual formulas for sine, cosine and tangent are not given as this level of study is that the computations are nightmarishly difficult.

You will be expected to memorize the values for sine, cosine, and tangent at some commonly used angles such as 30°, 45°, 60°, etc. There is a method for finding the values of sine and cosine for angles that are multiples of 3°, but it is quite tedious and takes a long time.

At this level of study, if you are not allowed a calculator, it is the custom either for you to be given a chart containing the values of the trigonometric functions you will need OR for you just to express your answer in terms of the trig function -- you might have an answer like `x = 17.4 sin 17.35°` or perhaps something like `θ=arcin (0.804)`
• I want to know how to calculate tan,sin,cos without calculator • I understand that the calculator is supposed to make it easier, but what exactly is the calculator doing? If I didn't have a calculator, what would I have to do? I find it difficult to understand math if I can't do it without a calculator. 