Question

If \(3 \sin \theta+4 \cos \theta=5\), then the value of \(3 \cos \theta-4 \sin \theta\) is equal to (a) 1 (b) 5 (c) 0 (d) \(-5\)

Short answer

Answer: (d) -5

Step-by-step solution

Divide the given equation by 5

Divide both sides of the equation by the constant 5, to get \(\frac{3}{5} \sin \theta + \frac{4}{5} \cos \theta = 1\). This will help us find the angle between the vectors of sine and cosine.

Square both sides of the equation and find the sum of both squares

We will square both sides of the equation obtained in step 1: \((\frac{3}{5} \sin \theta + \frac{4}{5} \cos \theta)^2 = 1^2\). Expanding this, we get: \((\frac{9}{25} \sin^2 \theta) + (\frac{24}{25} \sin \theta \cos \theta) + (\frac{16}{25} \cos^2 \theta) = 1\).

Simplify the equation

Since we know that \(\sin^2 \theta + \cos^2 \theta = 1\), we can simplify the equation as follows: \(\frac{1}{25} + \frac{24}{25} \sin \theta \cos \theta = 0\).

Find the angle

To find the angle in which the expression is valid, we can divide both sides of the simplified equation by \(\sin \theta \cos \theta\): \(\frac{1}{25} \cdot \frac{1}{\sin \theta \cos \theta} + \frac{24}{25} = 0\). Solving for \(\alpha\) in \(\tan \alpha = \frac{\sin \theta}{\cos \theta}\), we get: \(\alpha = \tan^{-1}(\frac{1}{24})\).

Find \(\sin \theta\) and \(\cos \theta\)

Using the angle \(\alpha\), we can find \(\sin \theta\) and \(\cos \theta\). \(\sin \theta = \frac{4}{5} \cos \alpha\), \(\cos \theta = \frac{3}{5} \cos \alpha\).

Substitute into the desired expression

Substitute \(\sin \theta\) and \(\cos \theta\) from Step 5 into the desired expression \(3 \cos \theta - 4 \sin \theta\): \(3 (\frac{3}{5} \cos \alpha) - 4 (\frac{4}{5} \cos \alpha) = -5\cos \alpha\).

Conclusion

The value of \(3 \cos \theta - 4 \sin \theta\) is equal to \(-5\cos \alpha\). Thus, the correct answer is (d) -5.

Key concepts

Sine and Cosine Relationships

Understanding the relationship between the sine and cosine functions is pivotal in solving trigonometric equations. The sine and cosine are defined as the y-coordinate and x-coordinate respectively of a point on the unit circle that makes an angle of \( \theta \) with the positive x-axis. Their relationship is often depicted in the trigonometric identity \( \sin^2 \theta + \cos^2 \theta = 1 \), known as the Pythagorean identity.

This fundamental relationship allows the manipulation of trigonometric equations to simplify and solve them. As they oscillate between -1 and 1, both sine and cosine are initially defined for angles in the first quadrant but are extended to all quadrants in a coordinate system through various signs.

When given an equation such as \( 3 \sin \theta + 4 \cos \theta = 5 \), one strategy is to notice the resemblance to the Pythagorean theorem. Dividing by 5, we aim to construct a right triangle where the legs have lengths proportional to 3 and 4, and the hypotenuse is 5. Thus, the scaled equation \( \frac{3}{5} \sin \theta + \frac{4}{5} \cos \theta = 1 \) can be viewed geometrically. This understanding is essential when finding the value of expressions like \( 3 \cos \theta - 4 \sin \theta \), which involves reversing the coefficients' roles to align with the same geometric interpretation.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the occurring angles. They serve as an indispensable tool for simplifying and solving trigonometric equations. One of the most basic and widely used identities is the Pythagorean identity mentioned earlier: \( \sin^2 \theta + \cos^2 \theta = 1 \).

This identity is a direct consequence of the Pythagorean theorem applied to a right-angled triangle within a unit circle. There are numerous other identities, such as angle sum and difference formulas, double angle formulas, and half angle formulas, which can transform and simplify complex trigonometric expressions. For instance, the given equation \( \frac{24}{25} \sin \theta \cos \theta \) from Step 3 of the solution can be addressed using these identities to simplify the problem further or find the desired angles. It's essential to be familiar with these to enhance problem-solving efficiency in trigonometry.

Problem-Solving in Trigonometry

Problem-solving in trigonometry involves a variety of techniques that include the strategic use of identities to simplify equations, geometric interpretations, and algebraic manipulations. A step-by-step approach helps in deciphering the problem systematically. To begin with, dissect the problem: look for known identities, patterns, or formulas that can be applied.

Once an expression is simplified, as in the exercise shown, it can often bring clarity to the problem's context, making the subsequent steps more intuitive. Next, isolate the trigonometric functions or angles to calculate their specific values or relationships. Primary strategies include squaring both sides to eliminate square roots or using identities to transform products into sums, as seen in the original solution's steps.

Finally, substitution is a powerful technique where the values obtained from prior steps are replaced back into other parts of the equation to find the desired result. This multi-step process ensures a methodical approach to finding solutions and builds foundational skills that are applicable across a wide array of trigonometry problems.