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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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Ch. 1 - Angles and the Trigonometric Functions
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Capitolo 1, Problema 3
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.
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Guida verificata passo dopo passo1
Recall that for a point \(P(x, y)\) on the unit circle corresponding to an angle \(t\), the coordinates are given by \(x = \cos(t)\) and \(y = \sin(t)\).
Identify the values of \(x\) and \(y\) from the point \(P\) on the unit circle. These values represent \(\cos(t)\) and \(\sin(t)\) respectively.
Use the definitions of the six trigonometric functions in terms of \(\sin(t)\) and \(\cos(t)\):
\(\sin(t) = y\),
\(\cos(t) = x\),
\(\tan(t) = \frac{y}{x}\) (provided \(x \neq 0\)),
\(\csc(t) = \frac{1}{y}\) (provided \(y \neq 0\)),
\(\sec(t) = \frac{1}{x}\) (provided \(x \neq 0\)),
\(\cot(t) = \frac{x}{y}\) (provided \(y \neq 0\)).
Substitute the values of \(x\) and \(y\) into these formulas to express each trigonometric function in terms of the coordinates of point \(P\).
Check the domain restrictions for each function to ensure the denominators are not zero, and interpret the results accordingly.
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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 (in radians) measured from the positive x-axis. The coordinates (x, y) represent the cosine and sine of t, respectively, which are fundamental for defining trigonometric functions.
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Percorso guidato
06:11
Introduction to the Unit Circle
Trigonometric Functions on the Unit Circle
On the unit circle, the primary trigonometric functions sine and cosine are given by the y- and x-coordinates of point P, respectively. Other functions like tangent, cotangent, secant, and cosecant can be derived from these values using their definitions as ratios, such as tangent = sine/cosine.
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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 x = cos(t) and y = sin(t) to compute sine and cosine, then calculate other functions like tangent (y/x), secant (1/x), cosecant (1/y), and cotangent (x/y), ensuring denominators are not zero.
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7:28Evaluate Composite Functions - Values Not on Unit Circle
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