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Course: Geometry (FL B.E.S.T.)>Unit 7

Lesson 5: 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.

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• 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.
• Sal mentioned in the video that mathematicians gave trignometric ratios names: sine, cosine, and tangent. But of all the names they could've picked in the world, why those three? I don't see any connection between the ratios and the names.
• Oddly enough, trigonometry is really about circles. And, as a result, the names for trig functions come from circles too.

Sine comes from a Sanskrit word meaning "chord". A chord is a segment joining two points on the circumference of a circle.

For cosine, the "co-" stands for complementary. Complementary angles are those that add to 90 degrees. If you take the cosine of an angle and the sine of its complement, you get the same answer. For example, cos(30)=sin(60).

When talking about circles, a tangent is a line that hits one point on a circle. We still use the word "tangent" (besides the trig function tan) today, especially in calculus.

There are three other trig functions that we don't use as often: cotangent, secant, and cosecant. A secant line is a line that intersects two points on a circle. Also, notice the pattern with the "co-". The complementary rule applies to tan/cot and sec/csc too.
• I feel like this is a dumb question, but what is theta?
• theta is a Greek letter which are commonly used for unknown angles, theta is one of the most common letter used
• 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?
• I think by "data", you meant "theta." Yes, the trigonometric ratios can be applied to all angles of a right triangle. When you have a non-right triangle, you will need to use some formulas to apply them.
• How do you determine the degree of an angle? Other than using a protractor! :D
• You can use a trig ratio and 2 of the known measurements, use the law of sines, law of cosines, etc. There are a bunch of ways to find the measure of an angle.
• Why is this in lesson 2? I really think this should be in lesson 1, since it teaches information that is used in lesson 1
• what is difference between similar and congruent triangles?
• If two triangles are congruent, they're exactly the same. However, if they are similar, they look like same but have different size.
• could someone explain what sal is talking about please I don’t quite understand . is there another video i should watch to understand this because i’m new to trigonometry so i haven’t got a clue what i’m doing 😆
• In this video, Sal is explaining how to determine the relationship between two right triangles when they share an angle of the same measure (theta). They explain that if two triangles have two angles in common, the third angle is also the same. Since the sum of the angles of any triangle is 180 degrees, this means that the two triangles are similar. He then goes on to explain that the ratio of corresponding sides of similar triangles is always the same. Using this fact, he derive several equations relating the sides of the two triangles. These equations are true for any right triangle with an angle theta and are the trigonometric functions.
• 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°)