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Find a positive angle less than 2𝜋 that is coterminal with 16𝜋 3
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Ch. 1 - Angles and the Trigonometric Functions
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Capitolo 1, Problema 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.
Guida verificata passo dopo passo1
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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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.
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Percorso guidato
06:11
Introduction to the Unit Circle
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.
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Percorso guidato
6:34Sine, Cosine, & Tangent on the Unit Circle
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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Percorso guidato
7:28Evaluate Composite Functions - Values Not on Unit Circle
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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.
<IMAGE>
sin 𝜋/6
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In Exercises 2–4, convert each angle in degrees to radians. Express your answer as a multiple of 𝜋. 315°
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