Question

In a right triangle, the acute angles have the relationship sin(x+12°)=cos(18°+2x)
sin
⁡
(
𝑥
+
12
°
)
=
cos
⁡
(
18
°
+
2
𝑥
)
. What is the value of x? What is the measure of the smaller angle? Use 1–2 sentences to explain how you arrived at your answers.

Answer 1

We start with the equation \(\sin(x + 12^\circ) = \cos(18^\circ + 2x)\). Using the co-function identity, \(\cos(\theta) = \sin(90^\circ - \theta)\), we can rewrite it as:

\[
\sin(x + 12^\circ) = \sin(72^\circ - 2x)
\]

This leads to two possible equations to solve:

1. \(x + 12^\circ = 72^\circ - 2x + k \cdot 180^\circ\), or
2. \(x + 12^\circ = 180^\circ - (72^\circ - 2x) + k \cdot 180^\circ\) (for some integer \(k\)).

For the first case where \(k = 0\):

\[
x + 2x = 72^\circ - 12^\circ
\]
\[
3x = 60^\circ \Rightarrow x = 20^\circ
\]

For the second case, simplifying gives us:

\[
x + 12^\circ = 108^\circ + 2x
\]
\[
12^\circ - 108^\circ = 2x - x \Rightarrow -96^\circ = x
\] (not valid for an angle x)

Thus, the only valid solution is \(x = 20^\circ\).

The smaller angle in the triangle is \(x\), which is \(20^\circ\).

I arrived at my answer by applying the co-function identity and simplifying the resulting equations to find the angle \(x\).