Question

 If $\sin 3\theta =\cos (\theta -2{}^\circ )\text{ where 3}\theta \text{ and }(\theta -2{}^\circ )$ are acute angles, what is the value of $\theta $?
a.22
b.23
c.24
d.25

Answer 1

Hint: Mathematics includes the study of topics which are related to quantity, structure, space and change. It has no generally accepted definition. It involves various operations which are performed using various operators such as addition multiplication division and subtraction. By the combination of these primary operators, trigonometric expression is obtained for a triangle. Using these properties, we can solve the above problem. The major expression for converting the problem into a single trigonometric variable is sin (90° - θ) = cos θ.

Complete step-by-step answer:

Mathematics is related to all the phenomena occurring in the world. When mathematical structures are good models of real phenomena mathematical reasoning can be used to provide insight or predictions about nature. One of the important sub branches of mathematics is trigonometry which involves the study of angles.

As per the given question, $\sin 3\theta =\cos (\theta -2{}^\circ )\text{ where 3}\theta \text{ and }(\theta -2{}^\circ )$ are acute angles:

Now, operating using the expression (90° - θ) = cos θ we get,
$\begin{align}
  & \sin 3\theta =\cos (\theta -2{}^\circ ) \\
 & \sin 3\theta =\sin (90-(\theta -2{}^\circ )) \\
 & \sin 3\theta =\sin (90+2-\theta ) \\
 & \sin 3\theta =\sin (92{}^\circ -\theta )\ldots (1) \\
\end{align}$

Now, for solving equation (1), we use the inverse property of sine transformation:
${{\sin }^{-1}}(\sin \theta )=\theta $ where θ is less than 90 in degrees.

Now, considering equation (1) and applying the inverse sine transformation we get,
$\begin{align}
  & {{\sin }^{-1}}(\sin 3\theta )={{\sin }^{-1}}(\sin (92{}^\circ -\theta )) \\
 & 3\theta =92{}^\circ -\theta \\
 & 4\theta =92{}^\circ \\
 & \theta =\dfrac{92{}^\circ }{4}=23{}^\circ \\
\end{align}$

Therefore, the value of θ is 23 in degrees.

Hence, option (b) is correct.

Note: The key step for solving this problem is transforming the two trigonometric functions into one trigonometric function. Once the same trigonometric function is present on the left-hand and right-hand side then evaluation for the angle can be done. This problem can also be solved alternatively by transforming the angle into cosine rather than sine.