Explain why sinx degrees= cos(90-x)degrees. include a diagram with your explanation.
construct a right angled triangle, let one non-90° angles be x, then the other is 90-x
recall sinx = opposite/hypotenuse
and cosx = adjacent /hypotenuse
but when we look at the angles x and 90-x, the opposite of one becomes the adjacent of the other
So it should be very easy to see why the relationship is true
or
using the identity
cos(A-B) = cosAcosB + sinAsinB
we have
cos(90-x) = cos90cosx + sin90sinx
= 0(cosx) + 1(sinx)
= sinx
To understand why sin(x) degrees = cos(90 - x) degrees, we need to delve into the trigonometric functions and the relationship between angles in a right triangle.
First, let's start with a diagram of a right triangle:
```
/|
/ |
/ |
/___|
```
In a right triangle, one angle is 90 degrees (denoted as 90°). We can label this angle as ∠B.
Now, let's label the other two angles as ∠A and ∠C. Since the sum of angles in a triangle is always 180 degrees, we know that ∠A + ∠C = 90 degrees.
```
/|
/A|
/ |
/___|
B
```
Next, let's focus on angles ∠A and ∠B. ∠A is the angle we are interested in for the sine function (sin(x) degrees), and ∠B is the angle we will explore for the cosine function (cos(90 - x) degrees).
The sine function is defined as the ratio of the length of the side opposite the angle (∠A) to the hypotenuse of the triangle. Let's label the hypotenuse as "h" and the side opposite ∠A as "a".
```
/|
/A| a
/ |
/___|
B
```
So, sin(x) degrees = a / h.
Now, let's look at the cosine function for the angle ∠B (cos(90 - x) degrees). The cosine function is defined as the ratio of the length of the adjacent side (to ∠B) to the hypotenuse (h). Let's label the adjacent side as "b".
```
/|
/A|
/ | a
/___|
b
```
Therefore, cos(90 - x) degrees = b / h.
But, notice that side "b" is actually the same as side "a". This is because the adjacent side to ∠B is the side opposite to ∠A in the same right triangle. So, we can replace "b" with "a" in the equation: cos(90 - x) degrees = a / h.
Now, comparing sin(x) degrees = a / h and cos(90 - x) degrees = a / h, we can see that they have the same ratio of "a / h". Therefore, sin(x) degrees = cos(90 - x) degrees.
So, in summary, sin(x) degrees is equal to cos(90 - x) degrees because they both represent the ratio of the side opposite the angle to the hypotenuse in a right triangle.