Hello!
Explain why the primary trigonometric ratios depend only on the given angle and not the size of legs and hypotenuse of a right triangle?
I am not 100% sure but it is because the angles have the same value for all of the ratios? Is it also because the side lengths increase/decrease proportionally as the size of triangle changes?
Thanks!
You should remember that two similar triangles have their corresponding angles equal, and their corresponding sides are in the same ratio
So if, for example, you have 2 right angled triangles, one of sides 3-4-5 and the other 6-8-10
the angles would be the same
so if we let their base angles be Ø in the first ..
sinØ = 4/5 and in the second sinØ = 8/10
but 8/10 = 4/5
so the ratio defined by sine stays constant
The same is true for the cosine and the tangent ratios.
your assumption is correct
Thanks! :)
Well, you're kind of on the right track, but let me put it in a slightly more "humorous" way for you:
Imagine you're at a party, and there's a really cool DJ playing some great tunes. The DJ looks at the dance floor and notices a group of people dancing in different spots, some closer to the DJ and some farther away.
Now, let's say the DJ wants to know how "in sync" these dancers are with the beat. In this case, the DJ represents the given angle, and the dancers represent the different sides of the triangle.
No matter where the dancers are on the dance floor, their "dance moves" (the primary trigonometric ratios) will depend only on the beat (the given angle), not their distance from the DJ.
So, it doesn't matter if the legs and the hypotenuse of a right triangle change in size. As long as the angle remains the same, the primary trigonometric ratios will always give you the same values, just like those dancers showing off their moves in sync with the DJ's music!
Hello!
You are partially correct in your explanation. The primary trigonometric ratios (sine, cosine, and tangent) indeed depend only on the given angle and not on the size of the legs and hypotenuse of a right triangle. Here's why:
The primary trigonometric ratios are defined as ratios of the lengths of the sides of a right triangle. Let's consider a right triangle with an acute angle, θ.
1. Sine (sin θ): The sine of an angle is defined as the ratio of the length of the side opposite the angle (opposite leg) to the length of the hypotenuse.
sin θ = opposite / hypotenuse
2. Cosine (cos θ): The cosine of an angle is defined as the ratio of the length of the side adjacent to the angle (adjacent leg) to the length of the hypotenuse.
cos θ = adjacent / hypotenuse
3. Tangent (tan θ): The tangent of an angle is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.
tan θ = opposite / adjacent
Now, let's analyze why these ratios depend only on the given angle and not on the size of the triangle:
1. Scaling: When you scale a right triangle (by multiplying the lengths of its sides by a constant factor), the ratios of the side lengths remain the same. This means that if you have two right triangles with the same acute angle θ, but different side lengths, their primary trigonometric ratios will be identical.
2. Similarity: Right triangles with the same acute angle θ are always similar. This means that their corresponding angles are equal, and their side lengths are proportional. As a result, the ratios of the corresponding side lengths (opposite, adjacent, and hypotenuse) will be the same.
Therefore, the primary trigonometric ratios depend solely on the given angle and remain constant regardless of the size of the legs and hypotenuse of the right triangle.