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Precalculus : Pre-Calculus

Study concepts, example questions & explanations for Precalculus

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12 Diagnostic Tests 380 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #2 : Solving Trigonometric Equations And Inequalities

If $\theta$ exists in the domain from $\left[0,\frac{\pi}{2}\right]$ , solve the following: $0=1-\cos^2\theta$

Possible Answers:

$0,\pi$

$0,\frac{\pi}{2}$

$\textup{No solution.}$

$\pi$

Correct answer:

Explanation:

Factorize $0=1-cos^2\theta$ .

$1-cos^2\theta= (1+cos(\theta))(1-cos(\theta))$

Set both terms equal to zero and solve.

$1+cos(\theta)=0$

$cos(\theta)=-1$

$\theta=\pi$

This value is not within the $[0,\frac{\pi}{2}]$ domain.

$1-cos(\theta)=0$

$cos(\theta)=1$

$\theta=0$

This is the only correct value in the $[0,\frac{\pi}{2}]$ domain.

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Example Question #4 : Solving Trigonometric Equations And Inequalities

Solve for $\theta$ in the equation $sin^2(\theta)-2sin(\theta)+1=0$ on the interval $0\leq\theta\leq2\pi$ .

Possible Answers:

$\theta=\frac{3\pi}{4}$

$\theta=sin^{-1}(1)$

$\theta=0$

$\theta=\frac{\pi}{2}$

$\theta=\frac{3\pi}{2}$

Correct answer:

$\theta=\frac{\pi}{2}$

Explanation:

If you substitute $x=sin(\theta)$ you obtain a recognizable quadratic equation which can be solved for ,

(x-1)^2=0 .

Then we can plug back into our equation and use the unit circle to find that

$\theta=\frac{\pi}{2}$ .

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Example Question #3 : Solving Trigonometric Equations And Inequalities

Given that theta exists from $[0,2\pi]$ , solve: $2cos^2\theta=cos(\theta)$

Possible Answers:

$\frac{\pi}{3}, \frac{5\pi}{3}$

$\textup{None of the given answers.}$

$\frac{\pi}{2},\frac{3\pi}{2}$

$\frac{\pi}{2}$

$\frac{\pi}{3},\frac{\pi}{2},\frac{3\pi}{2}, \frac{5\pi}{3}$

Correct answer:

$\frac{\pi}{3},\frac{\pi}{2},\frac{3\pi}{2}, \frac{5\pi}{3}$

Explanation:

In order to solve $2cos^2\theta=cos(\theta)$ appropriately, do not divide $cos(\theta)$ on both sides. The effect will eliminate one of the roots of this trig function.

Substract $cos(\theta)$ from both sides.

$2cos^2\theta-cos(\theta)=0$

Factor the left side of the equation.

$cos(\theta)[2cos(\theta)-1]=0$

Set each term equal to zero, and solve for theta with the restriction $[0,2\pi]$ .

$cos(\theta)=0$

$\theta = \frac{\pi}{2},\frac{3\pi}{2}$

$2cos(\theta)-1=0$

$2cos(\theta)=1$

$cos(\theta)=\frac{1}{2}$

$\theta= \frac{\pi}{3}, \frac{5\pi}{3}$

The correct answer is:

$\theta=\frac{\pi}{2},\frac{3\pi}{2},\frac{\pi}{3}, \frac{5\pi}{3}$

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Example Question #4 : Solving Trigonometric Equations And Inequalities

Solve $\sin^2(x) + 4 = 0$ for $0^{\circ}\leq x\leq 360^{\circ}$

Possible Answers:

There is no solution.

x = 0

x = -2, x = 2

$x = 90^{\circ}$

Correct answer:

There is no solution.

Explanation:

By subtracting from both sides of the original equation, we get $\sin^2(x) = -4$ . We know that the square of a trigonometric identity cannot be negative, regardless of the input, so there can be no solution.

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Example Question #2 : Solve Trigonometric Equations And Inequalities In Quadratic Form

Solve $\sin^2(x)-4 = 0$ when $0^{\circ}\leq x \leq 360^{\circ}$

Possible Answers:

x= 0

There are no solutions.

x=2, x=-2

$x = 45^{\circ}$

Correct answer:

There are no solutions.

Explanation:

Given that, for any input, $-1\leq \sin(x) \leq1$ , we know that $-1\leq\sin^2(x)\leq1$ , and so the equation $\sin^2(x)=4$ can have no solutions.

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Example Question #2 : Solve Trigonometric Equations And Inequalities In Quadratic Form

Solve $\sin^2(x)-1 =0$ when $0^{\circ}\leq x \leq 360^{\circ}$

Possible Answers:

$x = 90^{\circ}, x=270^{\circ}$

$x = 90^{\circ}$

There are no solutions.

$x = 0^{\circ}$

Correct answer:

$x = 90^{\circ}, x=270^{\circ}$

Explanation:

By adding one to both sides of the original equation, we get $\sin^2(x) = 1$ , and by taking the square root of both sides of this, we get $\sin(x) = 1, \sin(x) = -1.$ From there, we get that, on the given interval, the only solutions are $x = 90^{\circ}$ and $x = 270^{\circ}$ .

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Example Question #1 : Determine The Domain Of A Trigonometric Function

Which of the following is the correct domain of $cot(\theta)+3$ , where represents an integer?

Possible Answers:

$\textup{Everywhere except: }\pi k$

$\textup{Only Every: }2\pi k$

$\textup{Everywhere except: }2\pi k$

$\textup{Only Every: }\pi k$

$\textup{Everywhere except: }\frac{\pi}{2} k$

Correct answer:

$\textup{Everywhere except: }\pi k$

Explanation:

The cotangent graph only has a period of $\pi$ intervals and is most similar to the tangent graph. The domain of cotangent exists everywhere except every $\pi k$ value since an asymptote exists at those values in the domain.

The y-intercept of 3 shifts the cotangent graph up by three units, so this does not affect the domain.

Therefore, the graph exists everywhere except $\pi k$ , where is an integer.

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Example Question #2 : Trigonometric Functions

Please choose the best answer from the following choices.

What is the domain of the following function?

y=sin(x)

Possible Answers:

$\left ( 1,1\right )$

$\left [ 1,1\right ]$

$\left ( -\infty ,\infty \right )$

$\left [ -\infty ,\infty \right ]$

Correct answer:

$\left ( -\infty ,\infty \right )$

Explanation:

All x values make the function work. Thus, making the domain $\left ( -\infty ,\infty \right )$ . They're parentheses instead of brackets because parentheses are used when you can't actually use the specific value next to it. It is impossible to use infinity which makes parentheses appropriate. Brackets are used when you CAN use the specific value next to it.

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Example Question #1 : Trigonometric Functions

Please choose the best answer from the following choices.

What is the domain of the following function:

$y=\cos (x)$

Possible Answers:

$\left ( 0,1\right )$

$\left [ -\infty ,\infty \right ]$

$\left [ 0,1\right ]$

$\left ( -\infty ,\infty \right )$

Correct answer:

$\left ( -\infty ,\infty \right )$

Explanation:

All x values work for the function. Thus, making the domain all real numbers. Parentheses are required because you can never actually use the number infinity.

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Example Question #1 : Trigonometric Functions

Please choose the best answer from the following choices.

What is the domain of $y=\tan (x)$ .

Possible Answers:

$\left [ -\pi /2,\pi /2\right ]$

$\left ( -\pi /2,\pi /2\right )$

$\left \{ x\mid x\neq \pi /2+k\pi \right \}$

$\left ( -\infty ,\infty \right )$

Correct answer:

$\left \{ x\mid x\neq \pi /2+k\pi \right \}$

Explanation:

If you look at a graph of the function, you can see that every curve has a vertical asymptote that repeats every $\pi$ radians in the positive and negative x-direction, starting at $-\pi/2$ radians. Also, the curve has a length that stretches $\pi$ radians which makes the domain $\left \{ x\mid x\neq \pi /2+k\pi \right \}$ .

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Precalculus : Pre-Calculus

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #2 : Solving Trigonometric Equations And Inequalities

Example Question #4 : Solving Trigonometric Equations And Inequalities

Example Question #3 : Solving Trigonometric Equations And Inequalities

Example Question #4 : Solving Trigonometric Equations And Inequalities

Example Question #2 : Solve Trigonometric Equations And Inequalities In Quadratic Form

Example Question #2 : Solve Trigonometric Equations And Inequalities In Quadratic Form

Example Question #1 : Determine The Domain Of A Trigonometric Function

Example Question #2 : Trigonometric Functions

Example Question #1 : Trigonometric Functions

Example Question #1 : Trigonometric Functions

All Precalculus Resources

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