Question

Solve each equation for all nonnegative values of \(x\) less than \(360^{\circ} .\) Do some by calculator. $$1+\sin x=\sin x \cos x+\cos x$$

Short answer

Use a calculator to determine the nonnegative angles less than or equal to \(360^\circ\) for which \(\cos x = \frac{1}{\sin x + 1}\), ensuring \(\sin x eq -1\).

Step-by-step solution

Simplify the Equation

Subtract \(\sin x\) from both sides of the equation to get \(1 = \sin x \cos x + \cos x - \sin x\). Combine like terms to result in \(1 = \cos x ( \sin x + 1)\).

Isolate Cosine

Divide both sides of the equation by \(\sin x + 1\), assuming \(\sin x eq -1\), to isolate \(\cos x\). You get \(\frac{1}{\sin x + 1} = \cos x\).

Solve for x

Use a calculator to find the values of \(x\) that satisfy the equation \(\cos x = \frac{1}{\sin x + 1}\). Since the question asks for non-negative values of \(x\) less than \(360^\circ\), you must check the cosine values for \(x\) in the interval \[0, 360^\circ\].

Check for Excluded Values

Before finalizing your answer, double-check that \(\sin x eq -1\) in your solution since you divided by \(\sin x + 1\) earlier.

Key concepts

Solving Trigonometric Equations

To solve trigonometric equations, the goal is to isolate the variable (often denoted as \(x\)) and find all the angle values that make the equation true within a given interval. In the context of the given exercise, the approach involves a few strategic steps.

Firstly, manipulate the equation to simplify it, which may include combining like terms or factoring. For example, subtracting \(\sin x\) from both sides and then factoring out \(\cos x\) to simplify the equation.

Next, isolate the trigonometric function that contains the variable. This was done by dividing both sides of the equation by \(\sin x + 1\), assuming \(\sin x eq -1\).

Then, calculate angle measures typically using a calculator. The solutions must be within the required interval, here non-negative values less than \(360^\circ\), which corresponds to a full rotation in the unit circle.

Interpreting Calculator Results

When using a calculator, it's essential to understand the function's periodic nature. For cosine, this means the calculator may give one solution, but another solution is possible within the interval since \(\cos x\) has a period of \(360^\circ\).

Finally, check for excluded values, which are values that would make any denominator zero or invalidate the initial setup. This holistic approach to solving trigonometric equations ensures that all valid solutions are found.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the occurring variables. These identities are extraordinarily useful, as they allow for the simplification of trigonometric expressions, which in turn aids in solving trigonometric equations.

Common Identities

There are several basic identities that are widely used:

• Pythagorean Identity: \(\sin^2 x + \cos^2 x = 1\)
• Angle Sum and Difference Identities: Useful for expressing the sine or cosine of a sum or difference of angles in terms of the sine and cosine of each angle separately.
• Double-Angle Identities: They express trigonometric functions of double angles in terms of single angles.
• Reciprocal Identities: They relate reciprocals of trigonometric functions to other functions.

Understanding and applying these identities can transform a complex trigonometric equation into a more manageable form or simplify an expression to enable easier equation solving.

Inverse Trigonometric Functions

Inverse trigonometric functions are functions that reverse the action of the regular trigonometric functions. Where a trigonometric function takes an angle and gives a ratio, the inverse function takes a ratio and returns the possible angles that correspond to it.

Principal Values

The range of inverse trigonometric functions is restricted to their principal values so that they are functions in the mathematical sense (passing the vertical line test). For example, the principal value range for \(\arcsin x\) is \([-\frac{\pi}{2}, \frac{\pi}{2}]\) and for \(\arccos x\) is \([0, \pi]\).

Using inverse trigonometric functions is essential when solving a trigonometric equation that has been simplified to a basic form, such as \(\cos x = a\). To find the angle \(x\), one would use the inverse cosine function: \(x = \arccos(a)\). However, due to the periodic nature of trigonometric functions, there could be multiple angles that satisfy the equation within the given interval.

As a problem-solver, you must consider the domain and range of these functions and remember that additional solutions may exist based on the periodicity of the trigonometric functions.