Question

If \(x \tan 45^{\circ} \cdot \cos 60^{\circ}=\sin 60^{\circ} \cot 60^{\circ}\), then \(x\) is equal to 17 (a) 1 (b) \(1 / 2\) (c) \(\sqrt{3}\) (d) \(1 / \sqrt{2}\)

Short answer

a) 1 b) 1/2 c) sqrt(3)/2 d) sqrt(2)/2 Answer: b) 1/2

Step-by-step solution

Recall the given equation

We have the equation: $$x \tan 45^{\circ} \cdot \cos 60^{\circ} = \sin 60^{\circ} \cot 60^{\circ}$$ Step 2: Replace the trigonometric functions with their known values

Substitute the values of the trigonometric functions for special angles

We know that: \(\tan 45^{\circ} = 1\) \(\cos 60^{\circ} = \frac{1}{2}\) \(\sin 60^{\circ} = \frac{\sqrt{3}}{2}\) \(\cot 60^{\circ} = \frac{1}{\sqrt{3}}\) Now substitute these values into the equation: $$x \cdot 1 \cdot \frac{1}{2} = \frac{\sqrt{3}}{2} \cdot \frac{1}{\sqrt{3}}$$ Step 3: Simplify the equation

Simplify the equation by cancelling common factors

The equation simplifies to: $$\frac{1}{2}x = \frac{\sqrt{3}}{2} \cdot \frac{1}{\sqrt{3}}$$ Now cancel the common factor \(\frac{1}{2}\): $$x = \frac{\sqrt{3}}{2} \cdot \frac{1}{\sqrt{3}}$$ Step 4: Find the value of x

Calculate the value of x

Multiply the terms: $$x = \frac{\sqrt{3}}{2 \sqrt{3}}$$ Now cancel the common factor \(\sqrt{3}\): $$x = \frac{1}{2}$$ So the value of \(x\) is \(\frac{1}{2}\), which is answer (b).

Key concepts

Trigonometric Identities

Trigonometric identities are like handy tools in mathematics that allow us to simplify complex trigonometric expressions. These identities relate the six trigonometric functions - sine, cosine, tangent, cotangent, secant, and cosecant - to one another in various ways. In our exercise, we see identities at work through the use of tangent and cotangent functions, among others. Remember that:

• The tangent function, tan, at a 45-degree angle is always 1.
• The cotangent function, cot, at a 60-degree angle is the reciprocal of the tangent function at 60 degrees.

These known identities make it easier to substitute values and allow simplification. They act as the foundation upon which you can solve equations involving trigonometric functions efficiently.

Special Angles

Special angles in trigonometry often refer to 30°, 45°, and 60° angles. These angles have well-known sine, cosine, and tangent values that have been memorized due to their repetitive appearance in trigonometric problems. Here’s why this is important:

• At 45 degrees, the tangent function equals 1. This reflects the fact that the opposite and adjacent sides of a 45-degree angle in a right triangle are identical, as in an isosceles right triangle.
• At 60 degrees, the cosine function is 1/2, and the sine function is \(\sqrt{3}/2\).

Knowing these values by heart allows you to quickly replace trigonometric functions with their numerical equivalents, simplifying calculations considerably, as we did in our initial steps.

Simplification Techniques

When faced with trigonometric equations, simplifying them can often untangle complex problems into manageable parts. Here are some steps you can follow for simplification:

• Substitute trigonometric functions with known values for special angles, as seen in our solution.
• Cancel common factors on both sides of the equation. For example, in our original equation, dividing by \(\frac{1}{2}\) simplified the left-hand side, leading us towards the solution.
• After simplifying fractions, simplifying ratios of trigonometric functions can eliminate roots, helping to reveal the exact value for the variable like "x" in our problem.

With these techniques, even a seemingly complicated trigonometric equation becomes straightforward. It just takes some patience and practice to apply these systematically, leading to clarity and solution.