Domanda del libro di testo
Use the unit circle shown to find the value of the trigonometric function.
sin (2𝜋/3)
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Ch. 1 - Angles and the Trigonometric Functions
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Capitolo 1, Problema 2
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 point \(P(x, y)\) on the unit circle corresponds to the angle \(t\) (measured in radians) from the positive x-axis. Here, \(x = \cos(t)\) 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: \(\tan(t) = \frac{\sin(t)}{\cos(t)} = \frac{y}{x}\) (provided \(x \neq 0\)), \(\csc(t) = \frac{1}{\sin(t)} = \frac{1}{y}\) (provided \(y \neq 0\)), \(\sec(t) = \frac{1}{\cos(t)} = \frac{1}{x}\) (provided \(x \neq 0\)), and \(\cot(t) = \frac{1}{\tan(t)} = \frac{\cos(t)}{\sin(t)} = \frac{x}{y}\) (provided \(y \neq 0\)).
Summarize all the trigonometric function values at \(t\) using the coordinates of \(P(x, y)\) and ensure to check for any undefined values where denominators might be zero.
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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 measured from the positive x-axis, 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 at 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, and then find reciprocal functions if needed. This process links angle measures to function values.
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Percorso guidato
7:28Evaluate Composite Functions - Values Not on Unit Circle
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