Domanda del libro di testo
In Exercises 7–12, find the radian measure of the central angle of a circle of radius r that intercepts an arc of length s. Radius, r: 10 inches Arc Length, s: 40 inches
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Ch. 1 - Angles and the Trigonometric Functions
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Capitolo 1, Problema 6
In Exercises 1–8, a point on the terminal side of angle θ is given. Find the exact value of each of the six trigonometric functions of θ. (5, -5)
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Identify the coordinates of the point on the terminal side of angle \( \theta \). Here, the point is \( (5, -5) \), so \( x = 5 \) and \( y = -5 \).
Calculate the radius \( r \), which is the distance from the origin to the point, using the formula \( r = \sqrt{x^2 + y^2} \). Substitute the values to get \( r = \sqrt{5^2 + (-5)^2} \).
Recall the definitions of the six trigonometric functions in terms of \( x \), \( y \), and \( r \):
\[ \sin \theta = \frac{y}{r}, \quad \cos \theta = \frac{x}{r}, \quad \tan \theta = \frac{y}{x}, \quad \csc \theta = \frac{r}{y}, \quad \sec \theta = \frac{r}{x}, \quad \cot \theta = \frac{x}{y} \]
Substitute the values of \( x \), \( y \), and \( r \) into each of the six functions to express them exactly in terms of radicals and integers.
Simplify each expression if possible, keeping in mind the signs of \( x \), \( y \), and \( r \) to determine the correct sign of each trigonometric function.
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Coordinates and the Terminal Side of an Angle
The terminal side of an angle θ in standard position passes through a point (x, y). This point helps determine the angle's trigonometric values by relating x and y to the radius (r), which is the distance from the origin to the point, calculated using the Pythagorean theorem.
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Percorso guidato
05:32
Intro to Polar Coordinates
Definition of the Six Trigonometric Functions
The six trigonometric functions—sine, cosine, tangent, cosecant, secant, and cotangent—are ratios involving x, y, and r. Specifically, sin(θ) = y/r, cos(θ) = x/r, tan(θ) = y/x, and their reciprocals define the other three functions. These ratios are essential for finding exact values.
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6:04Introduction to Trigonometric Functions
Sign of Trigonometric Functions Based on Quadrants
The signs of x, y, and r determine the sign of each trigonometric function. Since r is always positive, the signs depend on the quadrant where the point lies. For (5, -5), the point is in the fourth quadrant, affecting the positivity or negativity of sine, cosine, and tangent values.
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6:04Introduction to Trigonometric Functions
Pratica correlata
Domanda del libro di testo
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.
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cos 5𝜋/6
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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. sin 𝜋/3
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Domanda del libro di testo
In Exercises 5–7, convert each angle in radians to degrees. 7𝜋 5
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Domanda del libro di testo
In Exercises 1–6, the measure of an angle is given. Classify the angle as acute, right, obtuse, or straight. 𝜋
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Domanda del libro di testo
In Exercises 1–6, the measure of an angle is given. Classify the angle as acute, right, obtuse, or straight. 𝜋/2
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