Find the Exact Value sec(-pi/3)-cot(-(5pi)/4)
The problem provided is asking for the calculation of the exact value of the expression involving the trigonometric functions secant (sec) and cotangent (cot), with the angles given in radians. Specifically, you need to determine the value of secant at negative pi over three, subtracted from the cotangent of negative five pi over four. This requires knowledge of the unit circle, the definitions and properties of trigonometric functions, and the ability to evaluate them at specific angles, understanding that negative angles indicate measurement in the clockwise direction from the positive x-axis.
$sec \left(\right. - \frac{\pi}{3} \left.\right) - cot \left(\right. - \frac{5 \pi}{4} \left.\right)$
Answer
Solution:
Step 1: Break down each trigonometric function.
Step 1.1
Rotate the angles by adding multiples of $2\pi$ to ensure they are within the range $0 \leq \theta < 2\pi$. This gives us $sec\left(\frac{5\pi}{3}\right) - cot\left(-\frac{\pi}{4}\right)$.
Step 1.2
Identify the corresponding acute angle for each function that has the same trigonometric value in the first quadrant. We get $sec\left(\frac{\pi}{3}\right) - cot\left(-\frac{\pi}{4}\right)$.
Step 1.3
Determine the exact value of $sec\left(\frac{\pi}{3}\right)$, which is $2$. This simplifies our expression to $2 - cot\left(-\frac{\pi}{4}\right)$.
Step 1.4
Again, rotate the angle by adding multiples of $2\pi$ to bring it within the desired range, resulting in $2 - cot\left(\frac{3\pi}{4}\right)$.
Step 1.5
Find the reference angle for the cotangent function and adjust the sign according to the quadrant. The result is $2 - (-cot\left(\frac{\pi}{4}\right))$.
Step 1.6
Compute the exact value of $cot\left(\frac{\pi}{4}\right)$, which is $1$. We now have $2 - (-1 \cdot 1)$.
Step 1.7
Simplify the multiplication of the negative signs.
Step 1.7.1
Calculate $-1$ times $1$, yielding $2 - (-1)$.
Step 1.7.2
Multiply $-1$ by $-1$ to get $2 + 1$.
Step 2: Combine the values.
Add $2$ and $1$ together to obtain the final result of $3$.
Knowledge Notes:
The problem involves finding the exact value of a trigonometric expression that contains secant and cotangent functions with negative angles. The process requires knowledge of trigonometric identities, reference angles, and the properties of trigonometric functions in different quadrants.
Trigonometric Functions: The secant (sec) and cotangent (cot) are reciprocal trigonometric functions. The secant of an angle is the reciprocal of the cosine, and the cotangent is the reciprocal of the tangent.
Reference Angles: A reference angle is the acute angle formed by the terminal side of an angle and the x-axis. For any angle in standard position, the reference angle has the same trigonometric values as the original angle but is always positive and lies in the first quadrant.
Negative Angles: Trigonometric functions of negative angles can be found by adding multiples of $2\pi$ (or $360^\circ$ for degrees) to the angle until it is positive. The trigonometric values of these angles are equivalent to those of their positive counterparts.
Quadrant Sign Rules: The sign of a trigonometric function depends on the quadrant in which the terminal side of the angle lies. For cotangent, it is positive in the first and third quadrants and negative in the second and fourth quadrants.
Exact Values: Certain angles, such as multiples of $\frac{\pi}{4}$ or $\frac{\pi}{6}$, have known exact trigonometric values that can be used to simplify expressions without the need for a calculator.
By applying these concepts, the problem-solving process involves converting negative angles to their positive equivalents, finding reference angles, determining the signs based on the quadrants, and using known exact values to simplify the expression to find the final result.
