Question

Solve each equation for \(0 \leq \theta < 2 \pi\) Give solutions to the nearest hundredth of a radian. a) \(\tan \theta=4.36\) b) \(\cos \theta=-0.19\) c) \(\sin \theta=0.91\) d) cot \(\theta=12.3\) e) \(\sec \theta=2.77\) f) \(\csc \theta=-1.57\)

Short answer

a) \(1.35, 4.49\) b) \(1.76, 4.52\) c) \(1.14, 2.00\) d) \(0.081, 3.22\) e) \(1.20, 5.08\) f) \(2.45, 5.59\)

Step-by-step solution

Understand the Problem

The task is to find the values of \(\theta\) that satisfy each trigonometric equation for \(\theta\) in the interval \(0 \leq \theta < 2\text{π}\). Each solution must be given to the nearest hundredth of a radian.

Solve for \(\theta\) in equation a

Given \(\tan \theta = 4.36\), use the inverse tangent function: \(\theta = \tan^{-1}(4.36)\). Calculate \(\theta\) using a calculator: \(\theta \approx 1.35\) radians. Since tangent is periodic with period \(\text{π}\), check within the interval \(0 \leq \theta < 2 \text{π}\): The solutions are \(\theta \approx 1.35\) and \(\theta \approx 1.35 + \text{π} \approx 4.49\) radians.

Solve for \(\theta\) in equation b

Given \(\text{cos } \theta = -0.19\), use the inverse cosine function: \(\theta = \text{cos}^{-1}(-0.19)\). Calculate \(\theta\) using a calculator: \(\theta \approx 1.76\) radians. Cosine is an even function, so the other solution in \(0 \leq \theta < 2 \text{π}\) is \(\theta = 2 \text{π} - 1.76 \approx 4.52 \) radians.

Solve for \(\theta\) in equation c

Given \(\text{sin } \theta = 0.91\), use the inverse sine function: \(\theta = \text{sin}^{-1}(0.91)\). Calculate \(\theta\) using a calculator: \(\theta \approx 1.14\) radians. Sine is periodic with period \(2 \text{π}\), the other solution in \(0 \leq \theta < 2 \text{π}\) is \(\theta \approx \text{π} - 1.14 \approx 2.00 \) radians.

Solve for \(\theta\) in equation d

Given \(\text{cot } \theta = 12.3\), which is the same as \(\frac{1}{\tan \theta} = 12.3\). First find \(\tan \theta = \frac{1}{12.3} \approx 0.081\). Now use the inverse tangent function: \(\theta = \tan^{-1}(0.081)\). Calculate \(\theta\) using a calculator: \(\theta \approx 0.081\) radians. The solutions are \( \theta \approx 0.081\) and \(\theta \approx 0.081 + \text{π} \approx 3.22 \) radians because tangent is periodic with period \( \text{π}\).

Solve for \(\theta\) in equation e

Given \(\text{sec } \theta = 2.77\), which is the same as \(\frac{1}{\text{cos } \theta} = 2.77\). First find \(\text{cos } \theta = \frac{1}{2.77} \approx 0.3617\). Use the inverse cosine function: \(\theta = \text{cos}^{-1}(0.3617)\). Calculate \(\theta\) using a calculator: \(\theta \approx 1.20 \) radians. Cosine is an even function, so the other solution in \(0 \leq \theta < 2 \text{π}\) is \(\theta = 2 \text{π} - 1.20 \approx 5.08 \) radians.

Solve for \(\theta\) in equation f

Given \(\text{csc } \theta = -1.57\), which is the same as \(\frac{1}{\text{sin } \theta} = -1.57\). First find \(\text{sin } \theta = \frac{1}{-1.57} \approx -0.637\). Use the inverse sine function: \(\theta = \text{sin}^{-1}(-0.637)\). Calculate \(\theta\) using a calculator: \(\theta \approx -0.69\) radians but adjust within the interval \(0 \leq \theta < 2 \text{π}\): \(\theta = -0.69 + 2\text{π} \approx 5.59 \) radians. The other solution in \(0 \leq \theta < 2 \text{π}\) is \(\text{π} - 0.69 \approx 2.45 \) radians.

Key concepts

tangent function

The tangent function, often abbreviated as \(\tan\), is one of the primary trigonometric functions. It is defined as the ratio of the opposite side to the adjacent side in a right-angled triangle. Mathematically, \(\tan \theta = \frac{\text{opposite}}{\text{adjacent}} \). The function is periodic with period \(\text{π}\), which means its values repeat every \(\text{π}\) radians. This is important when solving equations like \(\tan \theta = 4.36\), as you need to check for solutions within the interval \([0, 2 \text{π})\). Common uses of the tangent function include determining the slope of a line and solving problems involving angles of elevation and depression.

inverse trigonometric functions

Inverse trigonometric functions are used to find angles when given a trigonometric ratio. For example, if you know \(\tan \theta = 4.36\), you can find \(\theta\) by using the inverse tangent function, written as \(\theta = \tan^{-1}(4.36)\). Like other inverse functions, the primary output range is limited to ensure a unique value. For instance, \(\tan^{-1} x\) gives angles in the range \(-\frac{\text{π}}{2} \text{ to } \frac{\text{π}}{2}\). To find all possible solutions for \(\theta\) in equations, consider the periodic nature of trigonometric functions. In practice, use a calculator to find \(\theta\) and then apply the function's periodicity to find additional solutions within the desired interval.

radian measure

A radian is a unit of angular measure used in many areas of mathematics. One complete revolution around a circle is \(2 \text{π}\) radians. Unlike degrees, radians offer a direct relationship between the angle and arc length. For example, the length of the arc is equal to the angle in radians times the radius of the circle. When solving trigonometric equations within the interval \(0 \text{ to } 2 \text{π}\), it's often more convenient and accurate to use radian measure. Calculators typically handle trigonometric functions in radians more efficiently, making it easier to find solutions like \(1.35 \text{ radians and } 4.49 \text{ radians instead of degrees}\).

sine function

The sine function, denoted as \(\text{sin}\), is another fundamental trigonometric function. It represents the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle: \(\text{sin} \theta = \frac{\text{opposite}}{\text{hypotenuse}}\). The sine function is periodic with a period of \(2 \text{π}\), and its values range from \(-1 \text{ to } 1\). When solving \( \text{sin} \theta = 0.91 \), you first find one principal value using the inverse sine function: \( \theta = \text{sin}^{-1}(0.91) \), and then determine all other solutions within the interval by considering the periodicity and symmetry of the sine curve.

cosine function

The cosine function, represented as \(\text{cos}\), is similar to the sine function. It signifies the ratio of the adjacent side to the hypotenuse in a right-angled triangle: \(\text{cos} \theta = \frac{\text{adjacent}}{\text{hypotenuse}}\). The cosine function has a period of \(2 \text{π}\) and its range is from \(-1 \text{ to } 1\). Like the sine function, the cosine function is also even, meaning \( \text{cos}(-\theta) = \text{cos}(\theta) \). This property is useful when solving equations like \( \text{cos} \theta = -0.19 \) and helps identify that the other solution is \(\theta = 2 \text{π} - 1.76 \).