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Chapter 4: Problem 18

Use a sum or difference formula to find the exact value of the given trigonometric function. Do not use a calculator. $$ \tan 195^{\circ} $$

Short Answer

Expert verified
The exact value of \( \tan 195^{\circ} \) is \( 2 - \sqrt{3} \).

Step by step solution

01

Identify Angle Pair

To find \( \tan 195^{\circ} \), recognize that \( 195^{\circ} \) can be written as \( 180^{\circ} + 15^{\circ} \). This equation is suitable for a sum formula.
02

Use the Sum Formula

Use the tangent sum formula: \( \tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b} \). Here, \( a = 180^{\circ} \) and \( b = 15^{\circ} \).
03

Evaluate Tangent Values

Evaluate the tangent values: \( \tan 180^{\circ} = 0 \) and \( \tan 15^{\circ} = 2 - \sqrt{3} \). These are known values of tangent for special angles.
04

Substitute Values into the Formula

Substitute the values into the formula: \( \tan(180^{\circ} + 15^{\circ}) = \frac{\tan 180^{\circ} + \tan 15^{\circ}}{1 - \tan 180^{\circ} \tan 15^{\circ}} = \frac{0 + (2 - \sqrt{3})}{1 - 0 \cdot (2 - \sqrt{3})} \).
05

Simplify the Expression

Simplify the expression: \( \frac{2 - \sqrt{3}}{1} = 2 - \sqrt{3} \). Thus, the exact value is \( 2 - \sqrt{3} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Tangent Sum Formula
The tangent sum formula is a very important tool in trigonometry. It allows us to find the tangent of a sum or a difference of two angles. This formula is expressed as: \ \( \tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b} \). \ This formula is immensely helpful in simplifying complex angle computations, making it possible to find the value of a tangent function without a calculator. \
  • For example, when confronted with an angle like \( \tan 195^\circ \), the sum formula helps us by rewriting this as the sum \( 180^\circ + 15^\circ \).
  • By breaking down such an angle into familiar components, solving becomes a lot easier and requires less memorization of specific angle values.
The beauty of using formulas like these is that they show the interconnectedness of trigonometric functions and how understanding one angle can help in deducing others. \ Knowing how to apply this formula effectively is a skill that can simplify many trigonometric problems.
Special Angles
Special angles are specific degree measures that are commonly encountered in trigonometry. \ These include angles such as \( 0^\circ, 30^\circ, 45^\circ, 60^\circ, \text{and} \, 90^\circ \). \ These angles have known exact trigonometric values, making them very useful for calculations without a calculator. \
  • For instance, in solving for \( \tan 195^\circ \), we see that it's beneficial to use \( 180^\circ \), a special angle with the value of \( \tan 180^\circ = 0 \).
  • The angle \( 15^\circ \) itself is not a standard special angle, but its value, \( \tan 15^\circ = 2 - \sqrt{3} \), is derived from combinations of special angles.
Understanding these special angles and their trigonometric values allows for quick reference and faster computation, especially when dealing with complex expressions in trigonometric formulas.
Exact Trigonometric Values
Exact trigonometric values are those that can be precisely calculated without approximation. These include values for angles like \( 30^\circ \) where \( \sin 30^\circ = \frac{1}{2} \) or \( \cos 60^\circ = \frac{1}{2} \), and so on. \ Such values are essential in trigonometry as they form the basis for calculating other more complex angles. \
  • For example, \( \tan 15^\circ = 2 - \sqrt{3} \) is an exact value that can be very handy.
  • Almost every problem involving real-world applications relies on these exact values to ensure precision and accuracy.
By memorizing these exact values for the special angles, you can tackle many trigonometric problems with confidence and efficiency. \ Moreover, they simplify the otherwise tedious process of using approximations or calculators for every calculation, ensuring that your work is always on point.

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Most popular questions from this chapter

Find horizontally shifted sine and cosine functions so that each function satisfies the given conditions. Graph the functions. Amplitude 3 , period \(2 \pi / 3\), shifted by \(\pi / 3\) units to the right

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