Ch. 1 - Angles and the Trigonometric Functions

Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook

Blitzer 3rd Edition

Ch. 1 - Angles and the Trigonometric Functions

Problem 4

Chapter 1, Problem 4

In Exercises 1–4, a point P(x, y) is shown on the unit circle corresponding to a real number t. Find the values of the trigonometric functions at t.

Verified step by step guidance

Recall that the unit circle is a circle with radius 1 centered at the origin (0,0) in the coordinate plane. Any point P(x, y) on the unit circle satisfies the equation \(x^2 + y^2 = 1\).

Understand that for a real number \(t\), the coordinates of the point \(P(x, y)\) on the unit circle correspond to \(x = \cos(t)\) and \(y = \sin(t)\).

Use the coordinates of point \(P(x, y)\) to find the primary trigonometric functions: \(\sin(t) = y\) and \(\cos(t) = x\).

Calculate the other trigonometric functions using the definitions in terms of sine and cosine: \(\tan(t) = \frac{\sin(t)}{\cos(t)}\), \(\csc(t) = \frac{1}{\sin(t)}\), \(\sec(t) = \frac{1}{\cos(t)}\), and \(\cot(t) = \frac{\cos(t)}{\sin(t)}\).

Make sure to consider the signs of \(x\) and \(y\) based on the quadrant where point \(P\) lies to determine the correct signs of the trigonometric functions.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Unit Circle Definition

The unit circle is a circle with radius 1 centered at the origin of the coordinate plane. Each point P(x, y) on the unit circle corresponds to an angle t, where x = cos(t) and y = sin(t). This relationship allows us to define trigonometric functions based on coordinates.

Trigonometric Functions on the Unit Circle

The primary trigonometric functions—sine, cosine, and tangent—can be derived from the coordinates of point P on the unit circle. Specifically, sin(t) = y, cos(t) = x, and tan(t) = y/x (where x ≠ 0). Other functions like secant, cosecant, and cotangent are reciprocals of these.

Evaluating Trigonometric Functions for a Given Angle

To find the values of trigonometric functions at a real number t, identify the corresponding point P(x, y) on the unit circle. Use the coordinates to compute sine, cosine, and tangent values directly. This method simplifies evaluating trig functions without a calculator.

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Related Practice

Textbook Question

Find a positive angle less than 2𝜋 that is coterminal with 16𝜋 3

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Find the reference angle for 16𝜋 3

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In Exercises 1–6, the measure of an angle is given. Classify the angle as acute, right, obtuse, or straight. 87.177°

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In Exercises 1–8, a point on the terminal side of angle θ is given. Find the exact value of each of the six trigonometric functions of θ. (3, 7)

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Textbook Question

The unit circle has been divided into twelve equal arcs, corresponding to t-values of

0, 𝜋/6, 𝜋/3, 𝜋/2, 2𝜋/3, 5𝜋/6, 𝜋, 7𝜋/6, 4𝜋/3, 3𝜋/2, 5𝜋/3, 11𝜋/6, and 2𝜋

Use the (x,y) coordinates in the figure to find the value of each trigonometric function at the indicated real number, t, or state that the expression is undefined.

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sin 𝜋/6

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Textbook Question

In Exercises 2–4, convert each angle in degrees to radians. Express your answer as a multiple of 𝜋. 315°

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