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Ch. 3 - Radian Measure and The Unit Circle
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoLial 12th EditionCh. 3 - Radian Measure and The Unit CircleProblema 59
Capitolo 4, Problema 59

Without using a calculator, decide whether each function value is positive or negative. (Hint: Consider the radian measures of the quadrantal angles, and remember that π ≈ 3.14.)


tan 6.29

Guida verificata passo dopo passo
1
First, recognize that the angle given is in radians: 6.29 radians.
Recall that one full rotation around the unit circle is \(2\pi\) radians, which is approximately \(2 \times 3.14 = 6.28\) radians.
Since 6.29 radians is just a little more than \(2\pi\) radians, it means the angle has completed one full rotation and moved slightly beyond \(0\) radians (or \(0^\circ\)).
The tangent function has a period of \(\pi\) radians, so \(\tan(6.29)\) is equivalent to \(\tan(6.29 - 2\pi)\), which simplifies to \(\tan(6.29 - 6.28) = \tan(0.01)\) approximately.
Since \(\tan(0.01)\) is a very small positive angle in the first quadrant where tangent is positive, the value of \(\tan(6.29)\) is positive.

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Radian Measure and Quadrantal Angles

Radian measure expresses angles based on the radius of a circle, where 2π radians equal 360°. Quadrantal angles are multiples of π/2 (90°), such as 0, π/2, π, 3π/2, and 2π. Understanding where an angle lies relative to these helps determine the sign of trigonometric functions.
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Quadratic Formula

Periodic Nature of the Tangent Function

The tangent function has a period of π, meaning tan(θ) = tan(θ + nπ) for any integer n. This periodicity allows us to reduce large angles by subtracting multiples of π to find an equivalent angle within one period, simplifying the determination of the function's sign.
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Introduction to Tangent Graph

Sign of Tangent in Different Quadrants

Tangent is positive in the first and third quadrants and negative in the second and fourth. By locating the angle within the unit circle quadrants, one can decide the sign of tan(θ) without calculating its exact value.
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Sum and Difference of Tangent
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