Question

Make this shorter and like a 9th grader wrote it Based on the table provided, the sine and cosine have equal ratios when the angle is 45 degrees. At 45 degrees, both the sine and cosine are approximately 0.7071.

Conjecture: The sine and cosine of an angle are equal when the angle is 45 degrees. More generally, if
sin
⁡
(
𝑥
)
=
cos
⁡
(
𝑥
)
sin(x)=cos(x), then
𝑥
=
4
5
∘
+
18
0
∘
𝑛
x=45
∘
+180
∘
n for some integer
𝑛
n. However, within the range of 0 to 90 degrees, 45 degrees is the only solution. Also note that
sin
⁡
(
𝑥
)
=
cos
⁡
(
9
0
∘
−
𝑥
)
sin(x)=cos(90
∘
−x) so if
sin
⁡
(
𝑥
)
=
cos
⁡
(
𝑥
)
sin(x)=cos(x) then
𝑥
=
9
0
∘
−
𝑥
x=90
∘
−x which implies
2
𝑥
=
9
0
∘
2x=90
∘
and
𝑥
=
4
5
∘
x=45
∘
.

Answer 1

From the table, sin and cos are equal at 45°, and both ≈ 0.7071.

Conjecture: sin x = cos x when x = 45° + 180°n (n is any integer). In the range 0°–90°, the only solution is 45°.

Reason: sin x = cos(90° − x). If sin x = cos x, then x = 90° − x, so 2x = 90° and x = 45°.