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Chapter 5: Problem 7

Sketch an angle \(\theta\) in standard position such that \(\theta\) has the least posi. tive measure, and the given point is on the terminal side of \(\theta .\) Then find the values of the six trigonometric functions for each angle. Rationalize denominators when applicable. $$(7,-24)$$

Short Answer

Expert verified
The six trigonometric values are: \sin(\theta) = -\frac{24}{25}, \cos(\theta) = \frac{7}{25}, \tan(\theta) = -\frac{24}{7}, \csc(\theta) = -\frac{25}{24}, \sec(\theta) = \frac{25}{7}, \cot(\theta) = -\frac{7}{24}.

Step by step solution

01

Plot the Point

Start by plotting the point \(7, -24\) on the coordinate plane. This point will be on the terminal side of our angle \(\theta\) in standard position, where the vertex of the angle is at the origin \(0,0\), and the initial side is along the positive x-axis.
02

Calculate the Radius (r)

The radius \(r\) can be found using the Pythagorean theorem. Compute \(r\) as follows:\[ r = \sqrt{x^2 + y^2} = \sqrt{7^2 + (-24)^2} = \sqrt{49 + 576} = \sqrt{625} = 25 \]
03

Determine the Six Trigonometric Functions

Using the point \(7, -24\) and \(r=25\), we can determine the six trigonometric functions as follows:1. Sine (\text{sin} \theta): \[ \sin \theta = \frac{y}{r} = \frac{-24}{25} \]2. Cosine (\text{cos} \theta): \[ \cos \theta = \frac{x}{r} = \frac{7}{25} \]3. Tangent (\text{tan} \theta): \[ \tan \theta = \frac{y}{x} = \frac{-24}{7} \]4. Cosecant (\text{csc} \theta): \[ \csc \theta = \frac{r}{y} = \frac{25}{-24} = -\frac{25}{24} \]5. Secant (\text{sec} \theta): \[ \sec \theta = \frac{r}{x} = \frac{25}{7} \]6. Cotangent (\text{cot} \theta): \[ \cot \theta = \frac{x}{y} = \frac{7}{-24} = -\frac{7}{24} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Standard Position
The first step in understanding trigonometric problems involving angles is knowing what 'standard position' means. An angle is in standard position if its vertex is located at the origin of a coordinate plane and its initial side lies along the positive x-axis.
To sketch an angle in standard position, start from the positive x-axis and rotate the terminal side to the desired position. For the point (7, -24), you would plot this point in the fourth quadrant because both the x-coordinate is positive and the y-coordinate is negative.
Radius Using Pythagorean Theorem
To find the radius, also known as the hypotenuse in right-triangle terms, you can use the Pythagorean theorem. The theorem states: \( a^2 + b^2 = c^2 \) For our purposes, a and b are the x and y coordinates, and c is the radius (r).
In our example, substitute 7 for a and -24 for b to get: \[ r = \sqrt{7^2 + (-24)^2} = \sqrt{49 + 576} = \sqrt{625} = 25 \] This calculation shows that the radius is 25.
Six Trigonometric Functions
Using the coordinates of the given point and the radius, we can determine all six trigonometric functions. Here鈥檚 how:
  • **Sine (sin)**: \( \sin\ \theta = \frac{y}{r} = \frac{-24}{25} \)
  • **Cosine (cos)**: \( \cos\ \theta = \frac{x}{r} = \frac{7}{25} \)
  • **Tangent (tan)**: \( \tan\ \theta = \frac{y}{x} = \frac{-24}{7} \)
  • **Cosecant (csc)**: \( \csc\ \theta = \frac{r}{y} = \frac{25}{-24} = -\frac{25}{24} \)
  • **Secant (sec)**: \( \sec\ \theta = \frac{r}{x} = \frac{25}{7} \)
  • **Cotangent (cot)**: \( \cot\ \theta = \frac{x}{y} = \frac{7}{-24} = -\frac{7}{24} \)

These functions help describe the relationships between the angles and the lengths of the sides of a right triangle formed by the angle in the coordinate plane.
Rationalizing Denominators
When dealing with trigonometric functions, it's often necessary to rationalize denominators to make the expressions cleaner and easier to interpret. To rationalize a fraction, multiply both the numerator and denominator by the conjugate of the denominator or a suitable number that will eliminate any radicals.
For example, if you have a fraction like \( -\frac{25}{24} \) for cosecant, rationalization isn't needed because the denominator is already a rational number. However, in some cases with square roots or irrational numbers in the denominator, you would perform multiplication to adjust the fraction accordingly.
Coordinate Plane
The coordinate plane is a crucial concept for understanding trigonometric functions and angles in standard position. It consists of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical).
  • The origin is the point where these axes intersect \( (0,0) \)
  • The plane is divided into four quadrants where each has specific sign conventions for coordinates:

  • First Quadrant: \( x>0, y>0 \)
  • Second Quadrant: \( x<0, y>0 \)
  • Third Quadrant: \( x<0, y<0 \)
  • Fourth Quadrant: \( x>0, y<0 \)

Given the point (7, -24), it lies in the fourth quadrant. Knowing the quadrant helps determine the sign of the trigonometric functions.

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Most popular questions from this chapter

Give an expression that generates all angles coterminal with each angle. Let n represent any integer. $$0^{\circ}$$

When highway curves are designed, the outside of the curve is often slightly elevated or inclined above the inside of the curve. See the figure. This inclination is the superelevation. For safety reasons, it is important that both the curve's radius and superelevation be correct for a given speed limit. If an automobile is traveling at velocity \(V\) (in feet per second), the safe radius \(R\) for a curve with superelevation \(\theta\) is modeled by the formula $$R=\frac{V^{2}}{g(f+\tan \theta)}.$$ where \(f\) and \(g\) are constants. (Source: Mannering, \(\mathbf{F}\), and \(\mathbf{W}\). Kilareski, Principles of Highway Engineering and Traffic Analysis, Second Edition, John Wiley and Sons.) (a) A roadway is being designed for automobiles traveling at 45 mph. If \(\theta=3^{\circ}\) \(g=32.2,\) and \(f=0.14,\) calculate \(R\) to the nearest foot. (Hint: \(45 \mathrm{mph}=66 \mathrm{ft}\) per sec). (b) Determine the radius of the curve, to the nearest foot, if the speed in part (a) is increased to 70 mph. (c) How would increasing the angle \(\theta\) affect the results? Verify your answer by repeating parts (a) and (b) with \(\theta=4^{\circ}.\)

Decide whether each statement is possible or impossible for some angle \(\theta\). $$\cos \theta=-0.96$$

Find a value of \(\theta\) in the interval \(\left[0^{\circ}, 90^{\circ}\right]\) that satisfies each statement. Write each answer in decimal degrees to six decimal places as needed. See Example \(8 .\) $$\csc \theta=1.3861147$$

Give an expression that generates all angles coterminal with each angle. Let n represent any integer. $$-90^{\circ}$$
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