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In Exercises 61–86, use reference angles to find the exact value of each expression. Do not use a calculator. sin(-240°)
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Ch. 1 - Angles and the Trigonometric Functions
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Capitolo 1, Problema 1
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 quadrant of the angle \(t\) based on the signs of \(x\) and \(y\) to determine the signs of the trigonometric functions, ensuring the correct values for each function.
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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 find trigonometric function values directly from coordinates.
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Percorso guidato
06:11
Introduction to the Unit Circle
Trigonometric Functions on the Unit Circle
The six trigonometric functions—sine, cosine, tangent, cosecant, secant, and cotangent—can be defined using the coordinates of point P on the unit circle. Specifically, sin(t) = y, cos(t) = x, and tan(t) = y/x, with reciprocal functions defined accordingly.
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6:34Sine, Cosine, & Tangent on the Unit Circle
Sign of Trigonometric Functions in Quadrants
The sign of sine, cosine, and tangent depends on the quadrant where point P lies. For example, sine is positive in quadrants I and II, cosine is positive in quadrants I and IV, and tangent is positive in quadrants I and III. This helps determine the correct sign of function values.
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6:36Quadratic Formula
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