Question

If \(\cos \theta=\frac{4}{5}\) and \(0<\theta<90^{\circ}\), then the value of \(\frac{3 \cos \theta+2 \operatorname{cosec} \theta}{4 \sin \theta-\cot \theta}\) is : (a) \(-\frac{43}{2}\) (b) \(-\frac{41}{2}\) (c) \(\frac{43}{8}\) (d) \(-\frac{43}{6}\)

Short answer

Based on the step-by-step solution, the value of \(\frac{3 \cos\theta + 2 \operatorname{cosec}\theta}{4 \sin\theta - \cot\theta}\) is \(-\frac{43}{8}\) when \(\cos\theta = \frac{4}{5}\) and \(0 < \theta < 90^{\circ}\).

Step-by-step solution

Find the value of \(\sin\theta\)

Given that \(\cos \theta = \frac{4}{5}\), we can use the Pythagorean identity \(\sin^2\theta + \cos^2\theta = 1\) to find the value of \(\sin\theta\). Solve for \(\sin\theta\): \(\sin^2\theta = 1 - \cos^2\theta = 1 - \left(\frac{4}{5}\right)^2 = \frac{9}{25}\) Therefore, \(\sin\theta = \pm\frac{3}{5}\). Since \(0<\theta<90^{\circ}\), \(\sin\theta\) is positive. Thus, \(\sin\theta = \frac{3}{5}\).

Calculate the values of \(\operatorname{cosec}\theta\) and \(\cot\theta\)

Next, we will find the values of \(\operatorname{cosec}\theta\) and \(\cot\theta\). The cosecant function is the reciprocal of the sine function, so \(\operatorname{cosec}\theta = \frac{1}{\sin\theta}\). The cotangent function is the reciprocal of the tangent function, which is the quotient of the sine and cosine functions, so \(\cot\theta = \frac{\cos\theta}{\sin\theta}\). Using the values of \(\sin\theta\) and \(\cos\theta\) we found earlier, calculate the values of \(\operatorname{cosec}\theta\) and \(\cot\theta\): \(\operatorname{cosec}\theta = \frac{1}{\frac{3}{5}} = \frac{5}{3}\) \(\cot\theta = \frac{\frac{4}{5}}{\frac{3}{5}} = \frac{4}{3}\)

Substitute the values into the expression and simplify

Now, substitute the values of \(\cos\theta\), \(\operatorname{cosec}\theta\), and \(\cot\theta\) into the given expression: \(\frac{3 \cos \theta+2 \operatorname{cosec} \theta}{4 \sin \theta-\cot\theta} = \frac{3\left(\frac{4}{5}\right)+2\left(\frac{5}{3}\right)}{4\left(\frac{3}{5}\right)-\frac{4}{3}}\) Simplify the expression: \(= \frac{\frac{12}{5} + \frac{10}{3}}{\frac{12}{5} - \frac{4}{3}}\) \(= \frac{\frac{36+50}{15}}{\frac{36-20}{15}}\) \(= \frac{\frac{86}{15}}{\frac{16}{15}}\) Divide the numerators by the denominators: \(= -\frac{86}{16}\) Simplify the fraction: \(= -\frac{43}{8}\)

Choose the correct answer choice

Now, compare the simplified expression with the given answer choices. The correct answer is: (c) \(\frac{43}{8}\)

Key concepts

Pythagorean Identity

The Pythagorean Identity is a fundamental concept in trigonometry, which states that for any angle \( \theta \), this equation is always true: \ \[ \sin^2\theta + \cos^2\theta = 1 \] \ This identity is derived from the Pythagorean Theorem, hence its name. When given one trigonometric function, like cosine, you can easily find the sine using this identity. In our exercise, we know \( \cos \theta = \frac{4}{5} \). To find \( \sin \theta \), use the Pythagorean Identity: \ \[ \sin^2\theta = 1 - \cos^2\theta = 1 - \left( \frac{4}{5} \right)^2 = \frac{9}{25} \] \ Thus, \( \sin \theta = \pm\frac{3}{5} \). Since we are working in the first quadrant \( 0<\theta<90^{\circ} \), the value of \( \sin \theta \) is positive, hence \( \sin \theta = \frac{3}{5} \). Always remember that these trigonometric functions follow certain rules depending on the quadrant in which angle \( \theta \) resides.

Trigonometrical Ratios

Trigonometrical ratios are the basic building blocks of trigonometry. These ratios relate the angles of a triangle to the lengths of its sides. There are six primary trigonometric ratios: sine, cosine, tangent, cotangent, secant, and cosecant. In this exercise, we're primarily focused on: \

• The sine function \( \sin \theta \)
• The cosine function \( \cos \theta \)
• The quotient \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)

Each of these functions provides a unique ratio that can help solve various trigonometric problems. Understanding these ratios helps in calculations involving angles and sides of right triangles, and helps to form a basis for more complex mathematical concepts. Remember, in a right triangle:

• Sine is opposite over hypotenuse.
• Cosine is adjacent over hypotenuse.
• Tangent is opposite over adjacent.

Angle Measurement

Angle measurement is crucial in understanding trigonometry. Angles can be measured in degrees or radians. In our exercise, angle \( \theta \) is measured in degrees, specifically in the first quadrant where \( 0<\theta<90^{\circ} \). Different quadrants affect the calculations and signs of trigonometric functions:

• First quadrant \( (0^{\circ} - 90^{\circ}) \): All trigonometric functions are positive.
• Second quadrant \( (90^{\circ} - 180^{\circ}) \): Sine and cosecant are positive.
• Third quadrant \( (180^{\circ} - 270^{\circ}) \): Tangent and cotangent are positive.
• Fourth quadrant \( (270^{\circ} - 360^{\circ}) \): Cosine and secant are positive.

Understanding angles and their positions in each quadrant is essential to correctly applying the trigonometric ratios and their signs. Always visualize an angle on the unit circle or within a right triangle to grasp its measurement clearly.

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions are essential in trigonometry as they open up additional perspectives to solve problems. For every primary trigonometric function such as sine, cosine, and tangent, there is a reciprocal function:

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

In the exercise, we worked with the functions \( \operatorname{cosec} \theta \) and \( \cot \theta \). By understanding these relationships, you can find the reciprocal function values with the given primary functions. For instance, \( \operatorname{cosec} \theta \) is the reciprocal of \( \sin \theta \), and \( \cot \theta \) is the reciprocal of \( \tan \theta \), which is the quotient of \( \sin \theta \) and \( \cos \theta \). These reciprocals diversify problem-solving strategies and allow a comprehensive exploration of all possible solutions in trigonometric problems.