Question

Solve $$ \sin 2 \theta=-0.4010 \quad 0 \leq \theta \leq 2 \pi $$

Short answer

Question: Solve the trigonometric equation \(\sin 2\theta = -0.4010\) for \(\theta\) in the given range \(0 \leq \theta \leq 2 \pi\). Answer: The specific solutions for \(\theta\) in the given range are \(\theta_1 \approx 1.97103\) and \(\theta_2 \approx 1.17056\).

Step-by-step solution

Identify the general solutions for sin(2θ) = -0.4010

Recall the sine function range is between -1 and 1 inclusive, so the value -0.4010 is in range. To find the general solutions for the equation \(\sin 2 \theta = -0.4010\), we first find the inverse sine (also called arcsin) of -0.4010. $$ 2 \theta = \arcsin(-0.4010) $$ The inverse sine function has a range of \([-\pi/2, \pi/2]\), so the first solution \(2 \theta_1\) is in this range. To find the second solution \(2 \theta_2\), we use the symmetry of the sine function about the y-axis: $$ 2 \theta_2 = \pi - \arcsin(-0.4010) $$ Now, we have our two solutions for \(2\theta\).

Find the specific solutions for θ in the given range 0 ≤ θ ≤ 2π.

To find the specific solutions for \(\theta\) in the given range, we first need to divide both the general solutions for \(2\theta_1\) and \(2\theta_2\) by 2: $$ \theta_1 = \frac{1}{2} \arcsin(-0.4010), $$ $$ \theta_2 = \frac{1}{2} (\pi - \arcsin(-0.4010)). $$ Next, we add a multiple of \(\pi\) to \(\theta_2\) until it is in the range \(0 \leq \theta \leq 2\pi\). This is necessary because the sine function has periodicity \(2\pi\), and adding multiples of \(\pi\) will not change the value of \(\theta\) in the equation \(\sin 2\theta = -\,4010\). In this case, we find that the specific solutions for \(\theta\) in the given range are: $$ \theta_1 \approx 3.14159-1.97103=1.17056, $$ $$ \theta_2 \approx 1.97103, $$ Thus, the specific solutions for \(\theta\) in the given range are: \(\theta_1 \approx 1.97103\) and \(\theta_2 \approx 1.17056\).

Key concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are essential in solving equations that involve trigonometric functions like sine, cosine, and tangent. These functions allow us to find angles when given trigonometric values. For example, the inverse sine function, often written as \( \arcsin \), finds the angle whose sine is a specific value. The range of \( \arcsin \) is between \([ -\frac{\pi}{2}, \frac{\pi}{2}]\), meaning it will return an angle within this interval.
To solve the equation \( \sin 2\theta = -0.4010 \), we can use the inverse sine function. By calculating \( 2\theta = \arcsin(-0.4010) \), we find the first basic solution within the principal range of the inverse sine function. This step helps establish a primary angle, but because of the periodicity and symmetry properties of the sine function, other possible solutions may exist.
The key takeaway is that inverse trigonometric functions provide us a way to trace back from trigonometric values to angles. Understanding these allow us to solve more complex trigonometric equations.

Sine Function

The sine function is one of the primary trigonometric functions used to relate angles of a right triangle to the ratios of sides. Sine of an angle \(\theta\), denoted as \(\sin \theta\), gives the ratio of the length of the side opposite the angle to the hypotenuse in a right-angled triangle. The sine function is periodic and oscillates in a waveform between -1 and 1.
This property of sine is useful in recognizing that any value within this range is valid for sine, including -0.4010, as seen in our example problem. The function returns values repeating every \(2\pi\) interval, characteristic of its periodic nature.
Sine function's symmetry plays an important role: the graph is symmetric about the origin, which implies that \( \sin(\pi - \theta) = \sin(\theta) \) and \( \sin(2\pi - \theta) = -\sin(\theta) \), useful for finding additional solutions outside the principal range of arcsin. Understanding how sine functions behave helps solve equations involving these periodic oscillations effectively.

Periodicity in Trigonometry

Periodicity is a fundamental concept in trigonometry that describes how trigonometric functions repeat their values in a regular interval. For the sine function specifically, this period is \(2\pi\). This means that if you know the sine of an angle, you also know the sine of that angle plus any multiple of \(2\pi\).
In the context of the equation \(\sin 2\theta = -0.4010\), periodicity helps identify additional solutions by understanding that each valid angle can generate new angles simply by adding full cycles of the sine's period. For example, if you have one solution for \(2\theta\), say \(a\), all angles of the form \(a + 2k\pi\) where \(k\) is any integer, are also solutions.
Understanding periodicity is crucial as it assures that for every principal value, an infinite number of equivalent angles, spaced at regular intervals, exist. This allows us to comprehend solutions more holistically and effectively cover all possible answers in any given range.