Question

Use identities to find the exact value of each of the four remaining trigonometric functions of the acute angle \(\theta\). $$\sin \theta=\frac{\sqrt{3}}{2} \quad \cos \theta=\frac{1}{2}$$

Short answer

\(\tan \theta = \sqrt{3}\), \(\cot \theta = \frac{\sqrt{3}}{3}\), \(\sec \theta = 2\), \(\csc \theta = \frac{2\sqrt{3}}{3}\)

Step-by-step solution

- Identify Given Information

Given that \(\sin \theta = \frac{\sqrt{3}}{2}\) and \(\cos \theta = \frac{1}{2}\), these values lie in the first quadrant, which means \(\theta\) is an acute angle.

- Calculate \(\tan \theta\)

\(\tan \theta\) is defined as \(\frac{\sin \theta}{\cos \theta}\). Substitute the known values:\[\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \sqrt{3}\]

- Calculate \(\cot \theta\)

\(\cot \theta\) is the reciprocal of \(\tan \theta\). Thus:\(\cot \theta = \frac{1}{\tan \theta} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}\)

- Calculate \(\sec \theta\)

\(\sec \theta\) is the reciprocal of \(\cos \theta\). Substitute the known value:\(\sec \theta = \frac{1}{\cos \theta} = \frac{1}{\frac{1}{2}} = 2\)

- Calculate \(\csc \theta\)

\(\csc \theta\) is the reciprocal of \(\sin \theta\). Substitute the known value:\(\csc \theta = \frac{1}{\sin \theta} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3}\)

Key concepts

Sin

The sine function, denoted as \( \sin \theta \) \(, \), is one of the fundamental trigonometric functions. In a right-angled triangle, the sine of an angle \( \theta \) \( \) is defined as the ratio of the length of the opposite side to the length of the hypotenuse. For example, \( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \). In the exercise, we are given that \( \sin \theta = \frac{\sqrt{3}}{2} \). This value indicates that the ratio of the opposite side to the hypotenuse of angle \( \theta \) \( \) is \( \frac{\sqrt{3}}{2} \).To find the sine value, understanding its relationship within the triangle is crucial. Remember, if \( \theta \) \( \) is an acute angle (less than 90°), \( \sin \theta \) \( \) will always be a positive value.

Cos

The cosine function, written as \( \cos \theta \) \(, \), is another fundamental trigonometric function. In a right-angled triangle, \( \cos \theta \) \( \) is defined as the ratio of the length of the adjacent side to the hypotenuse:

• \( \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} \) \

In the exercise, we are given that \( \cos \theta = \frac{1}{2} \). This indicates that the adjacent side's length is half of the hypotenuse.

Tan

The tangent function, denoted as \( \tan \theta \), is the ratio of the sine of the angle to the cosine of the angle:

• \( \tan \theta = \frac{\sin \theta}{\cos \theta} \) \

Using the given values, we can calculate \( \tan \theta \) \( \) easily: \( \tan \theta = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \sqrt{3} \). This means the tangent of \( \theta \) \( \) is \( \sqrt{3} \).The tangent function is unique because it can be seen as a slope of the angle when looking at the unit circle.

Csc

The cosecant function, written as \( \csc \theta \), is the reciprocal of the sine function:

• \( \csc \theta = \frac{1}{\sin \theta} \) \

In the given exercise, substituting the known value \( \sin \theta = \frac{\sqrt{3}}{2} \) into the equation gives us: \( \csc \theta = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3} \). This means the cosecant function serves as a way to understand the inverse relationship to the sine function.

Sec

The secant function, represented as \( \sec \theta \) \(, \) is the reciprocal of the cosine function:

• \( \sec \theta = \frac{1}{\cos \theta} \).

According to the given information, \( \cos \theta = \frac{1}{2} \). This allows us to find \( \sec \theta \) \( \) as follows: \( \sec \theta = \frac{1}{\frac{1}{2}} = 2 \). The secant function can be helpful for understanding hyperbolic functions and is vital in advanced trigonometric identities.

Cot

The cotangent function is the reciprocal of the tangent function and is written as \( \cot \theta \):

• \( \cot \theta = \frac{1}{\tan \theta} \) \

Given that \( \tan \theta = \sqrt{3} \), we can find \( \cot \theta \) \( \) by calculating: \( \cot \theta = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3} \). The cotangent function is useful when dealing with complementary angles and finding slopes of lines perpendicular to \( \tan \theta \).

Acute Angle

An acute angle is an angle that is less than 90°. In trigonometry, angles in the first quadrant of the unit circle are all acute angles because they range between 0° and 90°.

• An important property of an acute angle is that all trigonometric functions of this angle will have positive values.

In our exercise, the given \( \sin \theta \) and \( \cos \theta \) values lie in this first quadrant where \( \theta \) is indeed less than 90°.

Reciprocal Identities

Reciprocal identities in trigonometry are very useful because they express the relationship between trigonometric functions and their reciprocals. These identities include:

• \( \csc \theta = \frac{1}{\sin \theta} \) \
• \( \sec \theta = \frac{1}{\cos \theta} \) \
• \( \cot \theta = \frac{1}{\tan \theta} \) \

In the provided exercise, we use these identities to find \( \csc \theta, \sec \theta, \) and \( \cot \theta \) from the given \( \sin \theta \) and \( \cos \theta \) values. Understanding reciprocal identities helps in connecting different trigonometric functions and simplifying trigonometric expressions.