Question

The value of \(\theta\) for which \(\sqrt{3} \cos \theta+\sin \theta=1\) is : (a) 0 (b) \(\pi / 3\) (c) \(\pi / 6\) (d) \(\pi / 2\)

Short answer

Answer: (b) \(\theta = \frac{\pi}{3}\)

Step-by-step solution

Substitute each choice in the equation

Let's substitute each choice in the equation and see which one makes the equation true.

Choice (a): \(\theta = 0\)

Substituting \(\theta = 0\) in the equation, we get: \(\sqrt{3} \cos(0) + \sin(0) = \sqrt{3} \cdot 1 + 0 = \sqrt{3} \neq 1\)

Choice (b): \(\theta = \frac{\pi}{3}\)

Substituting \(\theta = \frac{\pi}{3}\) in the equation, we get: \(\sqrt{3} \cos(\frac{\pi}{3}) + \sin(\frac{\pi}{3}) = \sqrt{3} \cdot \frac{1}{2} + \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{2} + \frac{\sqrt{3}}{2} = 1\)

Choice (c): \(\theta = \frac{\pi}{6}\)

Substituting \(\theta = \frac{\pi}{6}\) in the equation, we get: \(\sqrt{3} \cos(\frac{\pi}{6}) + \sin(\frac{\pi}{6}) = \sqrt{3} \cdot \frac{\sqrt{3}}{2} + \frac{1}{2} = \frac{3}{2} \neq 1\)

Choice (d): \(\theta = \frac{\pi}{2}\)

Substituting \(\theta = \frac{\pi}{2}\) in the equation, we get: \(\sqrt{3} \cos(\frac{\pi}{2}) + \sin(\frac{\pi}{2}) = \sqrt{3} \cdot 0 + 1 = 1 \neq 1\) By checking all the choices, we see that only choice (b) makes the equation true. Therefore, the answer is (b) \(\theta = \frac{\pi}{3}\).

Key concepts

Trigonometry

Trigonometry is an essential branch of mathematics focused on the relationships between the angles and sides of triangles, particularly right-angled triangles. One of its core functions is the study of sine (\text(sin)), cosine (\text(cos)), and tangent (\text(tan)), which are all based on the unit circle. With these trigonometric functions, we can solve equations and model periodic phenomena.

For instance, in the given exercise, the equation \(\sqrt{3} \cos \theta + \sin \theta = 1\) involves both the cosine and sine functions of the angle \(\theta\), presenting a classic example of a trigonometric equation. The equation can be understood as a linear combination of these two functions and can be solved by finding the specific angle(s) that satisfy this condition. Such problems are fundamental in various applications ranging from physics to engineering and computer graphics.

Problem Solving

Problem solving in mathematics is a systematic process that involves identifying and understanding the problem, devising a plan, carrying out the plan, and then looking back to check and interpret the results. The systematic approach taken in the step-by-step solution to the trigonometric equation exemplifies good problem-solving technique by exploring each option methodically.

The method of substituting potential solutions from the given choices and checking each one is a strategy known as trial-and-error or substitution, which is particularly effective for multiple-choice questions. This is a practical approach when the number of possible solutions is limited. It also provides a clear pathway to finding the correct answer, allowing students to verify each choice and understand why certain options do not work.

Radical Expressions

Radical expressions involve the use of roots, such as square roots \(\sqrt{x}\), cube roots \(\sqrt[3]{x}\), and higher-order roots. In the context of trigonometry, these expressions often appear in relation to the Pythagorean theorem, which relates the sides of a right triangle, or when simplifying trigonometric functions based on the unit circle properties.

In the exercise provided, the radical expression \(\sqrt{3}\) appears as a coefficient of the cosine function. Understanding how to manipulate and combine radical expressions with trigonometric functions is essential when solving trigonometric equations. Being comfortable with radical expressions allows for a seamless transition to solving more complex trigonometric equations that involve roots.

Angles in Radians

Angles can be measured in degrees or radians. In mathematics, especially calculus, angles are usually measured in radians because they provide a more natural measure in terms of the arc length of a circle. Radians are based on the radius of a circle, with one full revolution (360 degrees) equal to \(2\pi\) radians. A radian measures the angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle.

The angle choices in the given problem, such as \(\pi/3\), \(\pi/6\), and \(\pi/2\), are all in radians. It's essential to understand this unit of measurement because trigonometric functions in terms of radians often simplify to neater expressions than in degrees. For example, using radians, the well-known values of \(\sin(\pi/6)\) and \(\cos(\pi/3)\) can be quickly recognized and utilized to solve the question at hand.