Domanda del libro di testo
In Exercises 8–13, find the exact value of each expression. Do not use a calculator. sec 22𝜋 3
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Ch. 1 - Angles and the Trigonometric Functions
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Capitolo 1, Problema 10
In Exercises 5–18, the unit circle has been divided into twelve equal arcs, corresponding to t-values of 0, 𝜋, 𝜋, 𝜋, 2𝜋, 5𝜋, 𝜋, 7𝜋, 4𝜋, 3𝜋, 5𝜋, 11𝜋, and 2𝜋. 6 3 2 3 6 6 3 2 3 6 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.
tan 0
Guida verificata passo dopo passo1
Identify the angle t for which you want to find tan(t). Here, t = 0.
Recall that on the unit circle, the coordinates of a point corresponding to an angle t are given by (x, y) = (cos(t), sin(t)).
From the unit circle diagram, find the coordinates at t = 0. The coordinates are (1, 0).
Use the definition of the tangent function in terms of sine and cosine: \(\tan(t) = \frac{\sin(t)}{\cos(t)}\).
Substitute the values from the coordinates into the formula: \(\tan(0) = \frac{0}{1}\). Simplify this expression to find the value of tan(0).
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Unit Circle and Coordinates
The unit circle is a circle with radius 1 centered at the origin of the coordinate plane. Each point on the circle corresponds to an angle t measured in radians from the positive x-axis. The coordinates (x, y) of each point represent the cosine and sine of the angle t, respectively.
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Percorso guidato
06:11
Introduction to the Unit Circle
Trigonometric Functions on the Unit Circle
The sine, cosine, and tangent functions can be defined using the coordinates of points on the unit circle. For an angle t, cos(t) is the x-coordinate, sin(t) is the y-coordinate, and tan(t) is the ratio y/x, provided x ≠ 0. This allows evaluation of trig functions at various angles using the unit circle.
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6:34Sine, Cosine, & Tangent on the Unit Circle
Undefined Values of Tangent
Tangent is undefined when the cosine of the angle is zero because tan(t) = sin(t)/cos(t). On the unit circle, this occurs at points where the x-coordinate is zero, such as t = π/2 and 3π/2. Recognizing these points is essential to determine when tangent values do not exist.
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5:08Sine, Cosine, & Tangent of 30°, 45°, & 60°
Pratica correlata
Domanda del libro di testo
In Exercises 8–12, draw each angle in standard position. 8𝜋 3
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Domanda del libro di testo
In Exercises 5–18, the unit circle has been divided into twelve equal arcs, corresponding to t-values of
0, 𝜋, 𝜋, 𝜋, 2𝜋, 5𝜋, 𝜋, 7𝜋, 4𝜋, 3𝜋, 5𝜋, 11𝜋, and 2𝜋.
6 3 2 3 6 6 3 2 3 6
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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In Exercises 11–18, continue to refer to the figure at the bottom of the previous page.
csc 7𝜋/6
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Domanda del libro di testo
In Exercises 9–16, evaluate the trigonometric function at the quadrantal angle, or state that the expression is undefined. tan 𝜋
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Domanda del libro di testo
In Exercises 9–16, use the given triangles to evaluate each expression. If necessary, express the value without a square root in the denominator by rationalizing the denominator.
tan 30°
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Domanda del libro di testo
Use the given triangles to evaluate each expression. If necessary, express the value without a square root in the denominator by rationalizing the denominator.
<IMAGE>
sec 45°
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