Question

If tan \(\theta=2-\sqrt{3}\), then \(\tan (90-\theta)\) is equal to : (a) \(2+\sqrt{3}\) (b) \(2-\sqrt{3}\) (c) \(3+\sqrt{2}\) (d) \(3-\sqrt{2}\)

Short answer

(a) \(2+\sqrt{3}\)

Step-by-step solution

Given value

We are given that \(\tan \theta = 2 - \sqrt{3}\).

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

Recall that \(\cot\theta\) is the reciprocal of \(\tan\theta\). Therefore, we can find \(\cot\theta\) as follows: $$\cot\theta = \frac{1}{\tan\theta} = \frac{1}{2-\sqrt {3}}$$ Now, we rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator: $$\cot\theta = \frac{1}{2-\sqrt {3}} \times \frac{2+\sqrt {3}}{2+\sqrt {3}}$$ $$\cot\theta = \frac{2+\sqrt {3}}{(2-\sqrt {3})(2+\sqrt {3})}$$ Since \((a-b)(a+b) = a^2 - b^2\), we can simplify the denominator: $$\cot\theta = \frac{2+\sqrt {3}}{(2^2 - (\sqrt {3})^2)}$$ $$\cot\theta = \frac{2+\sqrt {3}}{4 - 3}$$ $$\cot\theta = 2+\sqrt {3}$$

Use the identity \(\tan(90 - x) = \cot x\) to find \(\tan(90-\theta)\)

Since we now have the value of \(\cot\theta\), we can use the identity \(\tan(90 - x) = \cot x\) to find \(\tan(90-\theta)\): $$\tan(90-\theta) = \cot\theta = 2+\sqrt {3}$$ So, the correct answer choice is: (a) \(2+\sqrt{3}\)

Key concepts

Tangent Function

The tangent function is one of the fundamental trigonometric functions defined for an angle in a right-angled triangle. It represents the ratio of the length of the opposite side to the length of the adjacent side. If we denote an angle by \(\theta\), then:

• \( \tan \theta = \frac{\text{Opposite}}{\text{Adjacent}} \)

This function is significant because it helps in solving problems involving angles and the lengths of sides in triangles.

Besides its geometric definition, the tangent function is also understood in the context of the unit circle. Here, it is the y-coordinate of the point where the terminal side of an angle intersects a line tangent to the circle at \((1,0)\). For all values of \(\theta\), tangent can be expressed as:

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

This relationship helps to find the tangent function values using sine and cosine, which are more straightforward in terms of coordinates on the unit circle.

Cotangent Function

The cotangent function is closely related to the tangent function and is defined as the reciprocal of the tangent function. For an angle \(\theta\), this means:

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

If the tangent of an angle is known, simply taking its reciprocal gives the cotangent.

In terms of a right-angled triangle, cotangent also has a simple interpretation. It represents the ratio of the length of the adjacent side to the length of the opposite side:

• \( \cot \theta = \frac{\text{Adjacent}}{\text{Opposite}} \)

This relationship shows the geometric significance of cotangent when solving problems involving side lengths and angles.

Another way to express cotangent is using sine and cosine:

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

This expression is particularly helpful when working within the unit circle framework of trigonometry and is used in deriving various trigonometric identities.

Trigonometric Identities

Trigonometric identities are mathematical equations that involve trigonometric functions and hold true for all values in their domains. They are vital tools in simplifying complex trigonometric expressions and solving equations.

One important identity involving the tangent and cotangent functions is:

• \( \tan(90^\circ - \theta) = \cot \theta \)

This identity is useful when transitioning between the functions as seen in our solution. It states that the tangent of the complement of an angle equals the cotangent of the angle itself.

Understanding identities like these allows for quick calculations without direct recourse to lengthy geometric reasoning.

Other fundamental trigonometric identities include:

• The Pythagorean identity: \( \sin^2 \theta + \cos^2 \theta = 1 \)
• Sum and difference formulas, such as \( \tan(a \pm b) = \frac{\tan a \pm \tan b}{1 \mp \tan a \tan b} \)

Trigonometric identities create a powerful framework aiding in the understanding and application of trigonometry across a wide range of problems.