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.

Answers

There are no human answers yet.
Answered by GPT-4o mini
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 \(x\).

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