Problem 64
Question
Find each exact value. Use a sum or difference identity. $$ \tan \left(-300^{\circ}\right) $$
Step-by-Step Solution
Verified Answer
\(\tan \left(-300^{\circ}\right) = \sqrt{3}\)
1Step 1: Convert the given angle's measurement to a standard form
The given angle is \(-300^{\circ}\). To convert it into a standard measurement, add \(360^{\circ}\) to it. Here, \(360^{\circ}\) is one complete revolution around the unit circle. Therefore, \(-300^{\circ} + 360^{\circ} = 60^{\circ}\). So, we have \(\tan \left(-300^{\circ}\right) = \tan (60^{\circ})\). This is because tangent function has a period of \(360^{\circ}\).
2Step 2: Calculate \(\tan (60^{\circ})\)
To calculate \(\tan (60^{\circ})\), recall the values of trigonometric ratios for standard angles. The exact value of \(\tan (60^{\circ})\) is equal to \(\sqrt{3}\).
3Step 3: Substitute the value of \(\tan (60^{\circ})\)
Finally, substitute the value of \(\tan (60^{\circ})\) into the equation from step 1 to get the exact value of \(\tan \left(-300^{\circ}\right)\). This gives us \(\tan \left(-300^{\circ}\right) = \sqrt{3}\).
Key Concepts
trigonometric identitiesangle conversionunit circleperiodicity of tangent
trigonometric identities
Trigonometric identities are equations involving trigonometric functions that are true for any value of the involved angles. These identities help simplify and solve trigonometric equations. Some of the most significant identities include Pythagorean identities, quotient identities, and angle sum or difference identities.
Pythagorean identities express one type of trigonometric function in terms of others. For example, the fundamental identity is \( \sin^2(\theta) + \cos^2(\theta) = 1 \). Other helpful identities are quotient identities, like \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \).
Angle sum and difference identities are vital because they allow calculations of trigonometric functions of combined angles, such as \( \tan(a \pm b) = \frac{\tan(a) \pm \tan(b)}{1 \mp \tan(a)\tan(b)} \). Understanding these relationships makes solving problems like finding \( \tan(-300^{\circ}) \) more manageable because you can convert angles and use identities effectively.
Pythagorean identities express one type of trigonometric function in terms of others. For example, the fundamental identity is \( \sin^2(\theta) + \cos^2(\theta) = 1 \). Other helpful identities are quotient identities, like \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \).
Angle sum and difference identities are vital because they allow calculations of trigonometric functions of combined angles, such as \( \tan(a \pm b) = \frac{\tan(a) \pm \tan(b)}{1 \mp \tan(a)\tan(b)} \). Understanding these relationships makes solving problems like finding \( \tan(-300^{\circ}) \) more manageable because you can convert angles and use identities effectively.
angle conversion
Angle conversion is the process of changing an angle measurement from one form to another, commonly between degrees and radians, or within a standard range like \(-180^{\circ}\) to \(180^{\circ}\). This task frequently occurs when solving trigonometric problems.
To convert between degrees and radians, use the conversion factors \(180^{\circ} = \pi \) radians. For negative angles, like \(-300^{\circ}\), it is often helpful to convert the angle by adding or subtracting full circles (\(360^{\circ}\)) to obtain a positive angle within a single revolution. Consequently, \(-300^{\circ} + 360^{\circ} = 60^{\circ}\), making calculations easier and more intuitive.
Remember, understanding how to convert and standardize angles is integral to applying trigonometric identities effectively.
To convert between degrees and radians, use the conversion factors \(180^{\circ} = \pi \) radians. For negative angles, like \(-300^{\circ}\), it is often helpful to convert the angle by adding or subtracting full circles (\(360^{\circ}\)) to obtain a positive angle within a single revolution. Consequently, \(-300^{\circ} + 360^{\circ} = 60^{\circ}\), making calculations easier and more intuitive.
Remember, understanding how to convert and standardize angles is integral to applying trigonometric identities effectively.
unit circle
The unit circle is a circle with a radius of one centered at the origin of a coordinate system. It is a convenient tool for understanding trigonometric functions because it visually represents angles and their corresponding coordinates, which relate to the cosine and sine values of those angles.
Each point on the unit circle corresponds to an angle measured from the positive x-axis. For instance, an angle of \(60^{\circ}\) corresponds to the point with coordinates \((\frac{1}{2}, \frac{\sqrt{3}}{2})\). The tangent function is the ratio \(\frac{\text{sine}}{\text{cosine}}\), therefore \( \tan(60^{\circ}) = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \sqrt{3} \).
By visualizing angles and their values with the unit circle, it simplifies tasks of determining exact function values and comprehending periodic behavior.
Each point on the unit circle corresponds to an angle measured from the positive x-axis. For instance, an angle of \(60^{\circ}\) corresponds to the point with coordinates \((\frac{1}{2}, \frac{\sqrt{3}}{2})\). The tangent function is the ratio \(\frac{\text{sine}}{\text{cosine}}\), therefore \( \tan(60^{\circ}) = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \sqrt{3} \).
By visualizing angles and their values with the unit circle, it simplifies tasks of determining exact function values and comprehending periodic behavior.
periodicity of tangent
The periodicity of tangent refers to the repeating pattern of the tangent function. The tangent function is periodic with a period of \(180^{\circ}\) or \(\pi\) radians, contrasting the \(360^{\circ}\) or \(2\pi\) radians period of the sine and cosine functions.
This shorter period means the tangent function repeats its values every \(180^{\circ}\). Therefore, \( \tan(\theta) = \tan(\theta + 180^{\circ}k) \) for any integer \(k\). Knowing the periodicity allows us to simplify problems by moving angles into a more convenient range.
In the example of \(\tan(-300^{\circ})\), recognizing its equivalency to \(\tan(60^{\circ})\) highlights how periodicity aids solving by reducing angles to familiar positions, enabling straightforward application of known trigonometric identities.
This shorter period means the tangent function repeats its values every \(180^{\circ}\). Therefore, \( \tan(\theta) = \tan(\theta + 180^{\circ}k) \) for any integer \(k\). Knowing the periodicity allows us to simplify problems by moving angles into a more convenient range.
In the example of \(\tan(-300^{\circ})\), recognizing its equivalency to \(\tan(60^{\circ})\) highlights how periodicity aids solving by reducing angles to familiar positions, enabling straightforward application of known trigonometric identities.
Other exercises in this chapter
Problem 63
Which equation is NOT true? \(\begin{array}{ll}{\mathbf{F} \cos ^{2} \theta=1-\sin ^{2} \theta} & {\text { G. } \cot ^{2} \theta=\csc ^{2} \theta-1} \\ {\text {
View solution Problem 64
Identify the period and tell where two asymptotes occur for each function. $$ y=\tan \frac{3 \pi \theta}{2} $$
View solution Problem 64
Use the fact that \(\frac{\pi}{6}=\frac{\pi}{2}-\frac{\pi}{3}\) to find an exact value for \(\cos \frac{\pi}{6} .\) Show your work.
View solution Problem 64
Graph each function in the interval from 0 to 2\(\pi\). \(y=\sin (x-\pi)+4\)
View solution