Problem 39
Question
Find the exact value of each expression, if possible. Do not use a calculator. $$ \tan ^{-1}\left[\tan \left(-\frac{\pi}{6}\right)\right] $$
Step-by-Step Solution
Verified Answer
The exact value of the expression \( \tan^{-1}[\tan(-\frac{π}{6})] \) is \(-\frac{π}{6}\).
1Step 1: Identify the Principal Interval
The principal interval for \( \tan^{-1}(x) \) is between \(-\frac{π}{2}\) and \(\frac{π}{2}\). This is the range of \( \tan^{-1}(x) \).
2Step 2: Check if Given Value is in Principal Interval
The given value is \(-\frac{π}{6}\). This falls within the principal interval of \(-\frac{π}{2}\) to \(\frac{π}{2}\). So, the inverse tangent of the tangent of \(-\frac{π}{6}\) will simply be \(-\frac{π}{6}\).
3Step 3: Simplify Expression
Since the given value is within the principal interval, the value of \( \tan^{-1}[\tan(-\frac{π}{6})] \) simplifies to \(-\frac{π}{6}\).
Key Concepts
Principal IntervalTan InverseExact Trigonometric ValuesSimplifying Trigonometric Expressions
Principal Interval
Understanding the concept of the principal interval is crucial when dealing with inverse trigonometric functions like the inverse tangent, or 'tan inverse'. The principal interval for 'tan inverse', denoted as tan−1(x), is the range of angles for which the function gives its primary set of output values.
For tan−1(x), this interval is between −π/2 and π/2. It’s important to note that this interval excludes the endpoints, meaning that tan−1(x) is undefined for π/2 or −π/2. The principal interval helps ensure that each input x corresponds to exactly one output in 'tan inverse', making the function reversible—hence the term 'inverse' trigonometric function.
Why is this useful? Whenever you're given a trigonometric expression involving the inverse tangent, checking whether the angle lies within the principal interval can guide you to the correct simplification.
For tan−1(x), this interval is between −π/2 and π/2. It’s important to note that this interval excludes the endpoints, meaning that tan−1(x) is undefined for π/2 or −π/2. The principal interval helps ensure that each input x corresponds to exactly one output in 'tan inverse', making the function reversible—hence the term 'inverse' trigonometric function.
Why is this useful? Whenever you're given a trigonometric expression involving the inverse tangent, checking whether the angle lies within the principal interval can guide you to the correct simplification.
Tan Inverse
'Tan inverse', also known as arctangent, denoted as tan−1 or arctan(x), refers to the inverse of the tangent function. This function is used to determine the angle whose tangent is the given number.
For instance, if tangent of angle A is x, then 'tan inverse' of x is angle A (assuming A is in the principal interval). In the exercise provided, this concept is applied to find the angle for the tangent of −π/6. Since −π/6 falls within the principal interval, we directly derive that the 'tan inverse' of tangent of −π/6 is simply −π/6.
For instance, if tangent of angle A is x, then 'tan inverse' of x is angle A (assuming A is in the principal interval). In the exercise provided, this concept is applied to find the angle for the tangent of −π/6. Since −π/6 falls within the principal interval, we directly derive that the 'tan inverse' of tangent of −π/6 is simply −π/6.
Example of Tan Inverse
If you have the value tangent of angle A equals 1, applying 'tan inverse' to 1 would give you the angle A, which is π/4, because tan(π/4) = 1.Exact Trigonometric Values
In the realm of trigonometry, there are certain angles for which the sine, cosine, and tangent values are exactly known without measurement; these are called exact trigonometric values. Recognizing and remembering these values is essential for simplifying trigonometric expressions effectively.
Common angles include 0, π/6, π/4, π/3, and π/2 radians (or their degree equivalents). For these angles, the trigonometric functions yield simple, rational numbers or square-root expressions. For example, the exact trigonometric value of tan(−π/6) is −1/√3.
Applying these known values helps in solving problems with precision and without a calculator. In the context of the given exercise, since the angle −π/6 is one for which an exact value is known, it facilitates the ease of finding the value of the 'tan inverse' expression.
Common angles include 0, π/6, π/4, π/3, and π/2 radians (or their degree equivalents). For these angles, the trigonometric functions yield simple, rational numbers or square-root expressions. For example, the exact trigonometric value of tan(−π/6) is −1/√3.
Applying these known values helps in solving problems with precision and without a calculator. In the context of the given exercise, since the angle −π/6 is one for which an exact value is known, it facilitates the ease of finding the value of the 'tan inverse' expression.
Simplifying Trigonometric Expressions
Simplifying trigonometric expressions involves a variety of techniques, including applying exact trigonometric values, using trigonometric identities, and considering principal intervals. The goal of simplification is to rewrite complex trigonometric expressions into more manageable and understandable forms.
For the inverse trigonometric functions, simplification typically means finding the associated angle in the principal interval that has the stated trigonometric value. In the given exercise, simplification was straightforward because the angle of −π/6 was within the principal interval for 'tan inverse'.
As a strategy, always check:
For the inverse trigonometric functions, simplification typically means finding the associated angle in the principal interval that has the stated trigonometric value. In the given exercise, simplification was straightforward because the angle of −π/6 was within the principal interval for 'tan inverse'.
As a strategy, always check:
- The principal interval to ensure the angle is within the correct range.
- Known exact trigonometric values for common angles.
- Relevant trigonometric identities that could simplify the expression further.
Other exercises in this chapter
Problem 38
find the reference angle for each angle. $$ 210^{\circ} $$
View solution Problem 38
In Exercises \(35-40,\) convert each angle in radians to degrees. Round to two decimal places. \(\frac{\pi}{17}\) radians
View solution Problem 39
In Exercises \(33-42,\) let $$ \sin t=a, \cos t=b, \text { and } \tan t=c $$ Write each expression in terms of \(a, b,\) and \(c .\) $$ \sin (-t-2 \pi)-\cos (-t
View solution Problem 39
An object moves in simple harmonic motion described by the given equation, where \(t\) is measured in seconds and \(d\) in inches. In each exercise, graph one p
View solution