What is the Sine Ratio?
Sine of an angle in a right triangle is the ratio between the length of the opposite side and the hypotenuse.
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Cosine of an angle in a right triangle is the ratio between the length of the adjacent side and the hypotenuse.
Here are some sine and cosine values for commonly used angles:
| Angle | Sine Value | Decimal Value | Cosine Value | Decimal Value |
|---|---|---|---|---|
Sine and cosine have many important applications in everyday life, especially in:
Let's look at an example of applying sine and cosine in the context of pyramids:
Suppose an archaeologist wants to know the height of a pyramid. They know that the elevation angle from the base to the top of the pyramid is and the slant height (edge) of the pyramid is .
To calculate the height of the pyramid, we use the sine ratio:
Therefore, the height of the pyramid is .
Now, if we want to know the base radius of the pyramid, we can use the cosine ratio:
Therefore, the base radius of the pyramid is .
All three are trigonometric ratios in right triangles.
All three change their values according to the angle.
These three ratios have a mathematical relationship:
A child is flying a kite and has managed to raise it to a height of . The child is holding the string at a height of from the ground. If the kite string forms an angle of with the ground, what is the length of the string being used?
To solve this problem, which trigonometric ratio should we use?
Correct Solution:
We need to calculate the string length (hypotenuse)
We know the effective height of the kite, which is .
We know the elevation angle .
Since we're looking for the hypotenuse and we know the opposite side (effective height), we use the sine ratio:
Therefore, the length of the kite string being used is approximately .