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## Get ready for AP® Calculus

### Course: Get ready for AP® Calculus>Unit 1

Lesson 11: Introduction to the trigonometric ratios

# Triangle similarity & the trigonometric ratios

Sal explains how the trigonometric ratios are derived from triangle similarity considerations. Created by Sal Khan.

## Want to join the conversation?

• What do you learn about a triangle from finding the sine, cosine and tangent? •  I wasnt sure that the other answers were really answering yours, they seemed to be more deep. If your question wasn't meant to be deep then I can answer it. When you take the sine, cosine, or tangent of a number you usually get a decimal number. Tangent is different, its usually a bigger number than the others. Now, this decimal number seems useless, i mean what do you do with it? Well, you can use this number to find a missing side length of a right triangle. Say you have all the angle measures but only one side length of a right triangle. You have the length of side A and you need to know the length of side B. Find which one you need sin, cos, or tan and enter it in the calculator. You get the answer so know you multiply the answer times the length of side A and the answer you get is side B!
• Can a cosine be negative? If so, when is it negative and when (if it can be positive) is it positive? Whenever I try to find cosines on my calculator, it is negative. Is that right? •   A cosine can be negative if the angle is more than 90 degrees and less than 270 degrees.
If you are using a calculator, you have to make sure it is set to degrees and not radians. If it is set to radians, you will get the wrong value all the time and you will sometimes get negatives when your answer should be positive.
A simple check to see if your calculator is right is to take cos of 60 degrees. The answer should be 0.5 , if the calculator thought it was 60 radians the answer will be -0.95241298
Hope this helps.
• Do the trigonometric definitions sine, cosine, and tangent apply to any angles of the right triangle? Meaning, can data be the 90 degree angle or can it only be one of the base angles? • How do you determine the degree of an angle? Other than using a protractor! :D • what is difference between similar and congruent triangles? • I feel like this is a dumb question, but what is theta? • Why are there not functions to calculate the ratios of angles other than 90°?

e.g.
sinₓ°(θ°) = opposite/hypotenuse of θ° in a x° triangle.
cosₓ°(θ°) = adjacent/hypotenuse of θ° in a x° triangle.
tanₓ°(θ°) = opposite/adjacent of θ° in a x° triangle.

Here we could define hypotenuse as the angle opposite to x°, opposite as the side opposite to θ° and adjacent as the side adjacent to θ° that is not the hypotenuse.

And this should work because of triangle similarity(Euclid's Elements, Book VI, Proposition 4):
angle 1 = x°
angle 2 = θ°
angle 3 = 180-x°-θ°

Establishing a relationship like this would help us solve for angles and sides in non-90° triangles. e.g.:
x° = 60°
θ° = 70°

side adjacent to 70° = x
side opposite to 70° = 5
tan₆₀°(70°) = 5/x
x = 5/tan₆₀°(70°)

Thank you • Law of Sines takes care of that.
sin(𝐴)∕𝑎 = sin(𝐵)∕𝑏 ⇒ 𝑎∕𝑏 = sin(𝐴)∕sin(𝐵)

In other words, the ratio between any two sides in any triangle is equal to the ratio between the sines of their opposite angles.

Given two angles, we easily calculate the third, and thereby we can find any trig ratio we want just using the sine function.

In your example, the angle opposite to side 𝑥 is 180° − (60° + 70°) = 50°, and so
5∕𝑥 = sin(70°)∕sin(50°) ⇒ 𝑥 = 5 sin(50°)∕sin(70°)
• How is Sine 39 degree = 0.6293...? How is the scientific calculator makes this calculation? • at what is an 'angle theta'? • I had problems with my draft. Couldn't figure out the length of my roof with just knowing the angle and the floor lenght.

40 minutes later - Math has never made me so happy and excited in my entire life! • Assuming your roof is one of those tapered/peak ones, then it is an iso. triangle. (Think of the flat part as the base.) You can divided that into two right triangles by cutting it straight thru the middle.

Since its cut in half (perpendicular bisector), its the floor length divided by 2.

If the angle is the one between the roof and the base, then its a cosine ratio (adjacent to hyp) to get the length of the floor.

cos(angle) = ((floor length)/2)/roof length

rearrange the equation:

roof length = (floor length)/cos(angle)
since its only one side of the roof, multiply it by two (iso. triangle to get the final result).