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Magazine Chapter 8: Problem 10
Express the given trigonometric function in terms of the same function of a positive acute angle. $$\tan 920^{\circ}, \csc \left(-550^{\circ}\right)$$
Short Answer
Expert verified
\[ \tan 920^{\circ} = \tan 20^{\circ} \] and \[ \csc(-550^{\circ}) = \csc 10^{\circ} \].
Step by step solution
01
Understand the Reference Angle for Tangent
To express \( \tan 920^{\circ} \) in terms of an acute angle, start by reducing the angle modulo 360 to find an equivalent angle between \(0^{\circ}\) and \(360^{\circ}\). Calculate \(920^{\circ} \mod 360^{\circ} \), which is the remainder when \(920^{\circ}\) is divided by \(360^{\circ}\). This gives \( \tan 920^{\circ} = \tan 200^{\circ} \).
02
Find the Acute Angle in the Tangent Function
The equivalent acute angle is found by reducing \(200^{\circ}\) to a reference angle in the unit circle's first quadrant. Since \(200^{\circ} > 180^{\circ}\), subtract \(180^{\circ}\): \(\theta = 200^{\circ} - 180^{\circ} = 20^{\circ}\). The tangent function has periodicity, so \( \tan 200^{\circ} = \tan 20^{\circ} \).
03
Understanding Reference for Cosecant
To handle \( \csc(-550^{\circ}) \), find an equivalent positive angle. Adding multiples of \(360^{\circ}\) can help. Calculate \(-550^{\circ} + 720^{\circ} = 170^{\circ}\) (adding twice the circle). Thus, \(\csc(-550^{\circ}) = \csc(170^{\circ})\).
04
Find the Acute Angle for Cosecant
Now, convert \(170^{\circ}\) into an acute angle. Since \(170^{\circ}\) is in the second quadrant, use \(180^{\circ} - 170^{\circ}\), which gives \(10^{\circ}\). The reciprocal identity for cosecant relates it to sine; hence, \( \csc(170^{\circ}) = \csc(10^{\circ}) \), because they share the same reference angle.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Reference Angle
In trigonometry, a reference angle is a helpful way to simplify the evaluation of trigonometric functions. It is always an acute angle, meaning it is between 0掳 and 90掳.
To find the reference angle when working with degrees:
To find the reference angle when working with degrees:
- First, identify the quadrant where the terminal side of the original angle lies after reducing it modulo 360掳 if necessary.
- If the angle is greater than 180掳 but less than 360掳, subtract 180掳 from it to find the reference angle.
Acute Angle
An acute angle is a key concept in trigonometry that represents an angle less than 90掳.
It is seen in many reference calculations for trigonometric functions.
When wanting to find a corresponding acute angle for a trigonometric function:
When wanting to find a corresponding acute angle for a trigonometric function:
- If dealing with an angle in the first quadrant, then that angle is already acute.
- In the second quadrant, subtract the angle from 180掳 to find the acute angle.
- In the third quadrant, subtract 180掳 from the angle.
- In the fourth quadrant, subtract the angle from 360掳.
Tangent Function
The tangent function, denoted as \( \tan \), relates the sides of a right triangle. It is defined as the ratio of the opposite side to the adjacent side. Mathematically, it is expressed as \( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \).
The function has important periodic properties:
The function has important periodic properties:
- Its period is 180掳, meaning \( \tan(\theta + 180掳) = \tan \theta \).
- The function returns to its original value every 180掳.
- This periodicity is crucial for simplifying the calculation of \( \tan 920^{\circ} \), which simplifies first to \( \tan 200^{\circ} \), and then to \( \tan 20^{\circ} \).
Cosecant Function
The cosecant function, denoted as \( \csc \), is the reciprocal of the sine function. Thus, \( \csc \theta = \frac{1}{\sin \theta} \).
This function is crucial in trigonometry, especially when dealing with non-standard angles.
This function is crucial in trigonometry, especially when dealing with non-standard angles.
- Cosecant is undefined where sine equals zero, corresponding to angles like 0掳, 180掳, and 360掳.
- The periodic nature of \( \csc \) is 360掳, meaning \( \csc(\theta + 360掳) = \csc \theta \).
- When working with negative angles like \( \csc (-550掳) \), it's necessary to transform them into a positive equivalent, leading us to \( \csc (170掳) \)..
