Textbook Question
In Exercises 49–59, find the exact value of each expression. Do not use a calculator. sin (22𝜋/3)
680
views
Ch. 1 - Angles and the Trigonometric Functions
Blitzer3rd EditionTrigonometryISBN: 9780137316601
Not the one you use?Change textbookSelect a different textbook
Table of contents
Chapter 1, Problem 1.RE.50
In Exercises 49–59, find the exact value of each expression. Do not use a calculator. tan 120°
Verified step by step guidance1
Recall that the tangent function has a period of 180°, so \( \tan(120^\circ) = \tan(120^\circ - 180^\circ) = \tan(-60^\circ) \), but it is often easier to use the reference angle in the second quadrant directly.
Identify the reference angle for 120°. Since 120° is in the second quadrant, the reference angle is \( 180^\circ - 120^\circ = 60^\circ \).
Recall the sign of tangent in the second quadrant. Tangent is negative in the second quadrant because sine is positive and cosine is negative, and \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \).
Use the exact value of \( \tan(60^\circ) \), which is \( \sqrt{3} \).
Combine the sign and the reference angle value to write \( \tan(120^\circ) = -\sqrt{3} \).
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
4mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Reference Angles
A reference angle is the acute angle formed between the terminal side of the given angle and the x-axis. For angles greater than 90°, finding the reference angle helps determine the trigonometric function's value by relating it to a known acute angle.
Recommended video:
5:31
Reference Angles on the Unit Circle
Signs of Trigonometric Functions in Quadrants
The sign of trigonometric functions depends on the quadrant in which the angle lies. For example, tangent is positive in the first and third quadrants and negative in the second and fourth. Knowing the quadrant of 120° helps determine the sign of tan 120°.
Recommended video:
6:36Quadratic Formula
Exact Values of Trigonometric Functions for Special Angles
Certain angles like 30°, 45°, 60° have known exact trigonometric values expressed in simplified radical form. Using these known values for the reference angle allows calculation of the exact value of tan 120° without a calculator.
Recommended video:
6:04Introduction to Trigonometric Functions
Related Practice
Textbook Question
In Exercises 49–59, find the exact value of each expression. Do not use a calculator. csc(-2𝜋/3)
648
views
Textbook Question
Use the unit circle shown to find the value of the trigonometric function.
sin (2𝜋/3)
692
views
Textbook Question
Use the unit circle shown to find the value of the trigonometric function.
tan 11𝜋/6
972
views
Textbook Question
In Exercises 49–59, find the exact value of each expression. Do not use a calculator. cot(-210°)
677
views
Textbook Question
The unit circle has been divided into eight equal arcs, corresponding to t-values of
0, 𝜋/4, 𝜋/2, 3𝜋/4, 𝜋, 5𝜋/4, 3𝜋/2, 7𝜋/4, and 2𝜋.
a. Use the (x,y) coordinates in the figure to find the value of the trigonometric function.
b. Use periodic properties and your answer from part (a) to find the value of the same trigonometric function at the indicated real number.
<IMAGE>
sin 3𝜋/4
604
views