Domanda del libro di testo
In Exercises 8–12, draw each angle in standard position. 5𝜋 6
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Ch. 1 - Angles and the Trigonometric Functions
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Capitolo 1, Problema 9
In Exercises 8–13, find the exact value of each expression. Do not use a calculator. tan 300°
Guida verificata passo dopo passo1
Recall that the tangent function has a period of 180°, so \( \tan(300^\circ) = \tan(300^\circ - 180^\circ) = \tan(120^\circ) \).
Identify the reference angle for 120°. Since 120° is in the second quadrant, the reference angle is \( 180^\circ - 120^\circ = 60^\circ \).
Determine the sign of tangent in the second quadrant. Tangent is negative in the second quadrant because sine is positive and cosine is negative, and \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).
Use the exact value of \( \tan 60^\circ \), which is \( \sqrt{3} \).
Combine the sign and the reference angle value to write \( \tan 300^\circ = -\sqrt{3} \).
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Reference Angles and Quadrants
Understanding reference angles helps simplify trigonometric values by relating them to acute angles. The angle 300° lies in the fourth quadrant, where tangent values are negative. Identifying the quadrant determines the sign of the trigonometric function.
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Percorso guidato
5:31
Reference Angles on the Unit Circle
Tangent Function Definition
The tangent of an angle in standard position is the ratio of the sine to the cosine of that angle (tan θ = sin θ / cos θ). Knowing this relationship allows calculation of tangent values using known sine and cosine values of reference angles.
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Percorso guidato
5:43Introduction to Tangent Graph
Exact Values of Special Angles
Certain angles like 30°, 45°, and 60° have well-known exact sine, cosine, and tangent values. Since 300° corresponds to 360° - 60°, using the exact values for 60° helps find the exact tangent of 300° without a calculator.
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Percorso guidato
04:3945-45-90 Triangles
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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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