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## Geometry (all content)

### Course: Geometry (all content)>Unit 13

Lesson 2: Introduction to the trigonometric ratios

# Trigonometric ratios in right triangles

Sal shows a few examples where he starts with the two legs of a right triangle and he finds the trig ratios of one of the acute angles. Created by Sal Khan.

## Want to join the conversation?

• what is trigonometry •   Study of triangles.
• At , why couldn't Sal just square the numerator and the denominator? Does this change the fraction in a way that multiplying does not? •   Yes. If you square the top and square the bottom of a fraction, you are probably multiplying the top and bottom by different numbers. This will change the overall value of the fraction. To rationalize a fraction (so that there is no radical in the denominator), you can't just square everything for this reason. You have to multiply the top and the bottom by the same number instead. (like square root of 64 over square root of 64, for example).
• How do you find the value of x when all you're given is the angle and the opposite side? • In terms of logic, it really depends on where x is. But if it's a side OTHER than the opposite side, you can solve it (if it's on the same triangle!). Okay it'll be hard to explain how to solve it here, you should really watch some other trig videos, but you can think of it this way, if you had a specific angle and a specific side, and that the triangle is right angled (that's the most important bit) then you know that another side must be a specific value or else the magnitude of other angles and sides will be changed. Simply think of it like this, you have a pyramid, you lock in two of the angles to the ground on the pyramid therefore the pyramid will not move at all like the side and the angles, which will mean it is one value that you can solve.
• What if you made theta the right angle? Then what's the adjacent side and what's the opposite? Is it possible to make theta the right angle? • How exactly do I describe how sin, cos, and tan are related? At the end of the video I saw that cos(60 degrees) would be the same as sin(30 degrees), and sin(60 degrees) would be the same as cos(30 degrees), and tangent(30 degrees) would be the inverse of tan(60 degrees). I see why -- the "opposite" of one angle would be the "adjacent" of the other, while the hypotenuse stays the same, so that's why they're reversed. Is there a more concise. general way to describe how they're related so I can understand without using the example triangle? • You can think of sin(θ) as cos(90°-θ). This is because, as doctorfoxphd said, the sine of one angle is the cosine of its compliment. That's actually why it's called co-sine, because it's the sine of the complimentary angle.

This is also the relationship between all the other cofunctions in trigonometry: tan(θ)=cot(90°-θ), sec=csc(90°-θ).

One other way to think about the relationship between a function and its cofunction is to think about the unit circle: your x-distance is described by cos(θ), and your y-distance described by sin(θ). But what if you turned the circle 90° and flipped it, and were measuring angles clockwise from the positive y-axis instead of counterclockwise from the positive x-axis? Now what was once the y-distance (sinθ) of a point on the circle is the x-distance of 90°-θ, or cos(90°-θ), and what was once the x-distance (cosθ) is now the y-distance of 90°-θ, or sin(90°-θ). Sine and cosine describe a circle, which is symmetric, in its relationship to a grid, so of course they look like one another.
• At around , Sal said that the functions are the inverse of each other. I understand what inverses are, since I've learned it in Algebra videos, but how can I tell that they're inverses? Is there any way to prove it? Thanks. • I've noticed in the proof for the Law of Cosines, the triangle is split-up into two right triangles. Because the Law of Cosines applies for all triangles, this would have to mean all triangles can also be split into two right triangles.

Does this property of triangles have a name in mathematics? • at sal said we cant simplify root 65 .but using factorization method we can find the exact square root of 65 that is 8.06 • Sin square A+cos square A=1,why??and is trigo useful anywhere??😰😒 • `sin²+cos² = 1`
`(O/H)² + (A/H)² = 1`
`(O²+A²)/H² = 1`
`O² + A² = H²`
This last line is the Pythagorean Theorem. So that's why. Read the lines from bottom to top for the proof starting from the Pythagorean Theorem.

Trig is useful in things such as physics and computer science. In computer science, trig is used to rotate elements on the screen (just for a simple example). There are other uses for it such as advanced sniper tactics. A sniper needs to estimate distances and adjust the angle of his aim based upon this. The relationship between sides of triangles and angles is trig, so a good sniper will do trig problems in his head to figure out the exact perfect way to aim. 