Question

Solve $$ \begin{gathered} \sin \theta \cos 41^{\circ}+\sin 41^{\circ} \cos \theta=0.6100 \\ 0^{\circ} \leq \theta \leq 360^{\circ} \end{gathered} $$

Short answer

Answer: To find the possible values of θ, first rewrite the equation using the angle addition formula for sine: sin(θ + 41°) = 0.6100. Then, find the principal solutions for θ by finding the inverse sine of 0.6100 and subtracting 41°. Next, find the general solutions for θ by considering the periodicity of the sine function and adding multiples of 360°. Finally, list all possible values of θ within the given domain by plugging these values into the general solution. The possible values of θ within the domain 0° ≤ θ ≤ 360° are: θ = arcsin(0.6100) - 41°, 180° - arcsin(0.6100) - 41°, and 360° - (arcsin(0.6100) + 41°).

Step-by-step solution

Rewrite the given equation using the angle addition formula

The given equation is: $$ \sin \theta \cos 41^{\circ}+\sin 41^{\circ} \cos \theta=0.6100 $$ Using the angle addition formula for sine function: $$ \sin (A+B) = \sin A \cos B + \cos A \sin B $$ We rewrite the given equation with A = θ and B = 41°: $$ \sin (\theta + 41^{\circ}) = 0.6100 $$

Solve the equation for θ

Now that we have the equation in terms of the sine function, we can find the principal solutions for θ by solving the following equation: $$ \sin (\theta + 41^{\circ}) = 0.6100 $$ To find the principal values of θ, we will first find the inverse sine of both sides: $$ \theta + 41^{\circ} = \arcsin(0.6100), \hspace{10mm} (*) $$ Once we find the values for these angles, we can subtract 41° from each value to find the principal solutions for θ.

Find the general solutions for θ

Now that we have the principal solutions for θ, let's consider more general solutions for θ by taking into account the periodicity of the sine function. The sine function has a period of 360°, meaning that: $$ \sin x = \sin y \hspace{5mm} \text{if} \hspace{5mm} x = y ± k \cdot 360^{\circ}, k \in \mathbb{Z} $$ Thus, the general solutions for θ will have the following form: $$ \theta = (\arcsin(0.6100)-41^{\circ} \pm k_1 \cdot 360^{\circ}) \ \text{or} \ (180^{\circ} - \arcsin(0.6100)-41^{\circ} \pm k_2 \cdot 360^{\circ}) \hspace{5mm} k_1, k_2 \in \mathbb{Z} $$

List all possible values of θ within the given domain

Now that we have the general solution, we need to find values of θ within the given domain (0° ≤ θ ≤ 360°). Plug the values into the general solution and list the values of θ that satisfy the domain: $$ θ = \arcsin(0.6100) - 41^{\circ}, \ \ 180^{\circ} - \arcsin(0.6100) - 41^{\circ}, \ 360^{\circ} - ( \arcsin(0.6100) + 41^{\circ}) $$ Calculate these values to obtain possible values of θ within the given domain. Keep in mind that while calculating, the arcsin(0.6100) should be calculated in degrees since the entire equation uses degree measures.

Key concepts

Angle Addition Formula

When you come across an equation involving sine and cosine terms, the angle addition formula can be your best friend. This formula helps simplify expressions and makes it easier to solve trigonometric equations. The angle addition formula for sine is:

• \( \sin(A + B) = \sin A \cos B + \cos A \sin B \)

In the context of our original problem, using this formula allows us to rewrite the equation \( \sin \theta \cos 41^{\circ} + \sin 41^{\circ} \cos \theta = 0.6100 \) into the simpler form \( \sin(\theta + 41^{\circ}) = 0.6100 \). By expressing the equation this way, it becomes more straightforward to find solutions for \( \theta \). This approach is especially useful when handling equations within a specific range, like 0° to 360°.

Sine Function

The sine function, a fundamental concept in trigonometry, captures the relationship between the size of an angle and the length ratios in a right triangle. When considering the equation \( \sin(\theta + 41^{\circ}) = 0.6100 \), the sine function represents the vertical position of the point on the unit circle. A sine value of 0.6100 might seem abstract, but it indicates a specific angle that relates back to trigonometric circle properties.
In trigonometry:

• Sine is periodic with a period of 360°.
• Values range from -1 to 1.
• Symmetric about the origin and repetitive every 360°.Understanding how sine values relate to angles is essential in solving and predicting angle measures.

Inverse Sine

The inverse sine function, denoted as \( \arcsin \), helps you find the angle whose sine value is known. In this specific task, \( \arcsin(0.6100) \) provides the principal angle whose sine equals 0.6100. However, remember that the sine function can yield the same value for multiple angles due to its periodic nature.
Here are key aspects of \( \arcsin \):

• The principal range is between -90° and 90°.
• For angles beyond this range, additional adjustments are necessary (like using complementary angles or considering function periodicity).

Taking the inverse will give you an initial angle, but you may need to consider adding or subtracting multiples of 360° to find all possible solutions within a specific set range, such as 0° to 360°.

Periodicity of Sine

Periodicity refers to the repeating nature of trigonometric functions, which is particularly evident with the sine function as it repeats every 360°. This characteristic is crucial when solving equations like \(\sin(\theta + 41^{\circ}) = 0.6100\).
When solving such equations, understanding how the periodicity affects the angle values can help you determine multiple solutions:

• \( \sin x = \sin y \) if \( x = y \pm k \cdot 360^{\circ}, k \in \mathbb{Z} \)
• It ensures that sine function solutions are distributed evenly across its period.
• Helps find both principal and additional solutions, keeping all solutions within a specified range like 0° to 360°.

By leveraging periodicity, you can ensure comprehensive and accurate solutions when handling trigonometric equations in circular domains.