Trigonometry Chapter 4: Trigonometric Functions of Angles
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Trigonometry Chapter 4: Trigonometric Functions of Angles

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Questions and Answers

What is the range of the sine and cosine functions?

[-1, 1]

How can you evaluate trigonometric functions for nonacute angles?

Using reference angles and reference right triangles

What is the formula for the tangent function in terms of x and y?

y/x (x ≠ 0)

What is the significance of the point (x, y) in defining trigonometric functions?

<p>It is a point on the terminal side of an angle θ in standard position</p> Signup and view all the answers

What is the relationship between the values of the sine and cosecant functions?

<p>They are reciprocals of each other</p> Signup and view all the answers

In which quadrants can the terminal side of an angle θ lie if tan θ is positive and cos θ is negative?

<p>II or III</p> Signup and view all the answers

What is the maximum value of the sine function?

<p>1</p> Signup and view all the answers

What is the relationship between the sine and cosecant functions?

<p>csc(u) = 1/sin(u)</p> Signup and view all the answers

In what quadrant would the terminal side of the angle lie if the cosine value is negative and the sine value is positive?

<p>II</p> Signup and view all the answers

What is the range of the secant function?

<p>All real numbers except -1 and 1</p> Signup and view all the answers

What is the period of the sine function?

<p>360°</p> Signup and view all the answers

What is the relationship between the tangent and cotangent functions?

<p>tan(u) = 1/cot(u)</p> Signup and view all the answers

What is the exact value of $u$ that satisfies the equation $sin(u) = 2$ in the interval $0 ≤ u ≤ 2π$?

<p>There is no solution in the interval $0 ≤ u ≤ 2π$ because $sin(u) ≤ 1$ for all values of $u$.</p> Signup and view all the answers

Find the exact values of $u$ that satisfy the equation $cos(u) = -1/3$ in the interval $0 ≤ u ≤ 2π$.

<p>The exact values of $u$ are $u = π + arcsin(1/3) ≈ 2.288$ and $u = 2π - arcsin(1/3) ≈ 3.852$.</p> Signup and view all the answers

What is the expected temperature on February 15 in Peoria, Illinois, according to the formula $T = 50 - 28 cos(2π(x - 31)/365)$?

<p>The expected temperature is $T = 50 - 28 cos(2π(46 - 31)/365) ≈ 24.4°F$.</p> Signup and view all the answers

Find the exact values of $u$ that satisfy the equation $tan(u) = 3$ in the interval $0 ≤ u ≤ 2π$.

<p>The exact values of $u$ are $u = arctan(3) ≈ 1.249$ and $u = π + arctan(3) ≈ 4.391$.</p> Signup and view all the answers

What is the expected temperature on August 15 in Peoria, Illinois, according to the formula $T = 50 - 28 cos(2π(x - 31)/365)$?

<p>The expected temperature is $T = 50 - 28 cos(2π(228 - 31)/365) ≈ 73.4°F$.</p> Signup and view all the answers

Find the exact values of $u$ that satisfy the equation $csc(u) = 2$ in the interval $0 ≤ u ≤ 2π$.

<p>The exact values of $u$ are $u = π/6$ and $u = 5π/6$.</p> Signup and view all the answers

If sin u is positive and cos u is negative, in which quadrant does the terminal side of u lie?

<p>Quadrant II</p> Signup and view all the answers

If tan u is negative and sec u is positive, in which quadrant does the terminal side of u lie?

<p>Quadrant IV</p> Signup and view all the answers

If cot u is positive and csc u is negative, in which quadrant does the terminal side of u lie?

<p>Quadrant I</p> Signup and view all the answers

If sin u = -a/b and the terminal side of u lies in Quadrant III, what is the value of cos u?

<p>-√(1 - (a/b)^2)</p> Signup and view all the answers

If tan u = a/b and the terminal side of u lies in Quadrant II, what is the value of sin u?

<p>-b/√(a^2 + b^2)</p> Signup and view all the answers

If sec u = -a/b and the terminal side of u lies in Quadrant III, what is the value of tan u?

<p>b/√(a^2 - b^2)</p> Signup and view all the answers

If the terminal side of an angle $u$ lies in Quadrant IV, what is the sign of its sine and cosine?

<p>The sine is negative, and the cosine is positive.</p> Signup and view all the answers

What is the relationship between the sine and cosine of an angle in Quadrant III and the same angle in Quadrant IV?

<p>They are the same, but with opposite signs.</p> Signup and view all the answers

How can you find the exact values of trigonometric functions using the unit circle?

<p>By using the coordinates of the terminal side of the angle on the unit circle.</p> Signup and view all the answers

If $
m{\sin} u = -0.3420$, and the terminal side of $u$ lies in Quadrant IV, what is the value of $
m{\cos} u$?

<p>approximately 0.9397</p> Signup and view all the answers

What is the difference between the trigonometric functions of an angle in Quadrant III and the same angle in Quadrant IV?

<p>The signs are opposite.</p> Signup and view all the answers

How can you use the fact that sine is an odd function and cosine is an even function to find the exact values of trigonometric functions?

<p>By using the properties of odd and even functions, such as $<br /> m{\sin}(-x) = -<br /> m{\sin}(x)$ and $<br /> m{\cos}(-x) = <br /> m{\cos}(x)$.</p> Signup and view all the answers

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