Question

Use the definition or identities to find the exact value of each of the remaining five trigonometric functions of the acute angle \(\theta\). $$\csc \theta=2$$

Short answer

sin θ = 1/2, cos θ = √3/2, tan θ = √3/3, csc θ = 2, sec θ = 2√3/3, cot θ = √3

Step-by-step solution

- Recall the definition of cosecant

The cosecant of an angle is the reciprocal of sine. That is, \(\text{csc } \theta = \frac{1}{\text{sin } \theta}\). Given \(\text{csc } \theta = 2\), we can find \(\text{sin } \theta\) using the formula: \(\text{sin } \theta = \frac{1}{\text{csc } \theta}\).

- Find sine

Using the given value \(\text{csc } \theta = 2\), we find \(\text{sin } \theta = \frac{1}{2}\).

- Use the Pythagorean identity

Using the Pythagorean identity \(\text{sin}^2 \theta + \text{cos}^2 \theta = 1\), we can find \(\text{cos } \theta\). Since \(\text{sin } \theta = \frac{1}{2}\), we have \((\frac{1}{2})^2 + \text{cos}^2 \theta = 1\), which simplifies to \(\text{cos}^2 \theta = \frac{3}{4}\).

- Find cosine

Taking the square root of both sides, we get \(\text{cos } \theta = \pm\sqrt{\frac{3}{4}} = \pm\frac{\sqrt{3}}{2}\). Since \(\theta\) is acute, we use the positive value: \(\text{cos } \theta = \frac{\sqrt{3}}{2}\).

- Calculate the tangent

The tangent of an angle is the ratio of sine to cosine. \(\text{tan } \theta = \frac{\text{sin } \theta}{\text{cos } \theta} = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}\).

- Find the secant

The secant of an angle is the reciprocal of cosine. \(\text{sec } \theta = \frac{1}{\text{cos } \theta} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3}\).

- Determine the cotangent

The cotangent of an angle is the reciprocal of tangent. \(\text{cot } \theta = \frac{1}{\text{tan } \theta} = \frac{1}{\frac{\sqrt{3}}{3}} = \sqrt{3}\).

- List the trigonometric functions

Now, summarize all the six trigonometric functions: \(\text{sin } \theta = \frac{1}{2}\), \(\text{cos } \theta = \frac{\sqrt{3}}{2}\), \(\text{tan } \theta = \frac{\sqrt{3}}{3}\), \(\text{csc } \theta = 2\), \(\text{sec } \theta = \frac{2\sqrt{3}}{3}\), and \(\text{cot } \theta = \sqrt{3}\).

Key concepts

cosecant

The cosecant function, abbreviated as \(\text{csc} \theta\), is a trigonometric function that is the reciprocal of the sine function. Its formula is \(\text{csc} \theta = \frac{1}{\text{sin} \theta}\). In this exercise, we were given \(\text{csc} \theta = 2\). By knowing that cosecant is the reciprocal of sine, we can find the sine value. If \(\text{csc} \theta = 2\), we solve for sine by finding \(\text{sin} \theta = \frac{1}{2} \). Understanding cosecant helps in finding missing trigonometric functions by using known identities.

sine

The sine function, denoted as \(\text{sin} \theta\), is one of the primary trigonometric functions representing the ratio of the length of the opposite side to the hypotenuse in a right triangle. For an acute angle \(\theta\), it’s essential to know that it ranges between 0 and 1. In our exercise, given \(\text{csc} \theta = 2\), we calculated \(\text{sin} \theta = \frac{1}{2} \). Once this value is found, it helps in applying further trigonometric identities to find other related functions like cosine and tangent.

Pythagorean identity

The Pythagorean identity is a fundamental relation among the trigonometric functions sine and cosine. It states \(\text{sin}^2 \theta + \text{cos}^2 \theta = 1\). This identity is very useful when one of the functions is known. In our exercise, \(\text{sin} \theta \) was found to be \(\frac{1}{2}\). Using the Pythagorean identity, we rearranged the equation to find \(\text{cos}^2 \theta\), leading to \(\text{cos}^2 \theta = \frac{3}{4}\). Solving for \(\text{cos} \theta \) then provides both positive and negative square roots, but only the positive value is taken for the acute angle.

cosine

The cosine function, represented as \(\text{cos} \theta\), denotes the ratio of the adjacent side to the hypotenuse in a right triangle. From the Pythagorean identity, once \(\text{sin} \theta\) is known, \(\text{cos} \theta\) can be determined. In this exercise, \(\text{cos}^2 \theta\) was found to be \(\frac{3}{4}\), thus \(\text{cos} \theta = \frac{\text{sqrt}{3}}{2}\). For angles between 0 and 90 degrees (acute), the cosine value is positive. Understanding cosine allows the calculation of other trigonometric functions like tangent and secant.

tangent

The tangent function, \(\text{tan} \theta\), is the ratio of sine to cosine functions: \(\text{tan} \theta = \frac{\text{sin} \theta}{\text{cos} \theta}\). In our solved exercise, knowing that \(\text{sin} \theta = \frac{1}{2}\) and \(\text{cos} \theta = \frac{\text{sqrt}{3}}{2}\), we computed \(\text{tan} \theta = \frac{\frac{1}{2}}{\frac{\frac{\text{sqrt}3}{2}}} = \frac{1}{\text{sqrt}3} = \frac{\text{sqrt}3}{3}\). The tangent function indicates the ratio of the opposite to the adjacent side in a right-angle triangle.

secant

Secant, abbreviated as \(\text{sec} \theta\), is the reciprocal of the cosine function: \(\text{sec} \theta = \frac{1}{\text{cos} \theta}\). From our exercise, \(\text{cos} \theta = \frac{\text{sqrt}3}{2}\), thus \(\text{sec} \theta = \frac{2}{\text{sqrt}3} = \frac{2\text{sqrt}3}{3}\). This makes the calculations straightforward once cosine is known. Secant function values help in various applications, such as signal analysis and physics.

cotangent

The cotangent function, denoted as \(\text{cot} \theta\), is the reciprocal of the tangent function: \(\text{cot} \theta = \frac{1}{\text{tan} \theta}\). In the given exercise, with \(\text{tan} \theta = \frac{\text{sqrt}3}{3}\), we determined \(\text{cot} \theta = \frac{1}{\frac{\text{sqrt}3}{3}} = \text{sqrt}3\). The cotangent function helps in determining the ratio of the adjacent to the opposite side in a right-angle triangle, encapsulating the inverse relationship of the tangent function.