Question

The expression \(4 \cos \left(\theta+\frac{\pi}{3}\right)+3 \sin \left(\theta+\frac{\pi}{3}\right)\) lies between (a) \(-5\) and 5 (b) \(-4\) and 4 (c) \(-3\) and 3 (d) \(-7\) and 7

Short answer

Answer: (d) -7 and 7

Step-by-step solution

Rewrite the expression using the angle addition formula

We will rewrite the expression using the angle addition formula for sine and cosine. The angle addition formulas are: \(sin(a + b) = sin(a)cos(b) + cos(a)sin(b)\) \(cos(a + b) = cos(a)cos(b) - sin(a)sin(b)\) Our expression is \(4 \cos \left(\theta+\frac{\pi}{3}\right)+3 \sin \left(\theta+\frac{\pi}{3}\right)\). So, we can rewrite the expression as: \(4 \left( \cos(\theta)\cos(\frac{\pi}{3}) - \sin(\theta)\sin(\frac{\pi}{3}) \right) + 3 \left( \sin(\theta)\cos(\frac{\pi}{3}) + \cos(\theta)\sin(\frac{\pi}{3}) \right)\)

Simplify the expression

Now, we will simplify the expression using the values of sine and cosine at \(\frac{\pi}{3}\). \(\sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2}\) and \(\cos(\frac{\pi}{3}) = \frac{1}{2}\) Thus, the expression becomes: \(4 \left( \cos(\theta) \cdot \frac{1}{2} - \sin(\theta) \cdot \frac{\sqrt{3}}{2} \right) + 3 \left( \sin(\theta) \cdot \frac{1}{2} + \cos(\theta) \cdot \frac{\sqrt{3}}{2} \right)\) \(2 \cos(\theta) - 2\sqrt{3} \sin(\theta) + \frac{3}{2} \sin(\theta) + \frac{3\sqrt{3}}{2} \cos(\theta)\)

Group terms

We will group the sine and cosine terms together: \(\left( 2 - 2\sqrt{3} \right) \sin(\theta) + \left( 2 + \frac{3\sqrt{3}}{2} \right) \cos(\theta)\)

Determine the range of values

Now, let's find the maximum and minimum value of the expression by considering the range of sine and cosine: \(\sin(\theta)\) and \(\cos(\theta)\) are bounded between -1 and 1. So, the minimum value of \(\left( 2 - 2\sqrt{3} \right) \sin(\theta)\) is \(-|2 - 2\sqrt{3}| = -2\sqrt{3} + 2\) and the maximum value is \(2\sqrt{3} - 2\). Similarly, the minimum value of \(\left( 2 + \frac{3\sqrt{3}}{2} \right) \cos(\theta)\) is \(-|2 + \frac{3\sqrt{3}}{2}| = -2 - \frac{3\sqrt{3}}{2}\) and the maximum value is \(2 + \frac{3\sqrt{3}}{2}\). The minimum value of the expression is thus \((-2\sqrt{3} + 2) + (-2 - \frac{3\sqrt{3}}{2}) = -3 - \frac{7\sqrt{3}}{2} \approx -7.05\). The maximum value of the expression is \((2\sqrt{3} - 2) + (2 + \frac{3\sqrt{3}}{2}) = 3 + \frac{7\sqrt{3}}{2} \approx 7.05\).

Compare with given options

Now, let's compare the range with the given options: (a) \(-5\) and 5 (b) \(-4\) and 4 (c) \(-3\) and 3 (d) \(-7\) and 7 The correct range is between approximately -7.05 to 7.05 which is closest to option (d) \(-7\) and 7. Answer: (d) \(-7\) and 7

Key concepts

angle addition formulas

To begin with, understanding how to use the angle addition formulas is crucial when working with trigonometric expressions. These formulas allow you to express the trigonometric functions of a sum of two angles in terms of the functions of the individual angles. This is particularly helpful when solving problems where you need to adjust an angle, such as addition or subtraction, to use known values of sine and cosine.

For any two angles, say \(a\) and \(b\):

• The angle addition formula for sine is \(\sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b)\).
• For cosine, it is \(\cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b)\).

Using these formulas effectively can help in transforming an otherwise complex trigonometric equation into something more manageable.

sine and cosine values

When simplifying trigonometric expressions, knowing specific values of sine and cosine for common angles is invaluable. This is because these values help to simplify the calculations involved in such problems. For example, when a problem involves angles like \(\frac{\pi}{3}\), it's handy to recall that:

• \(\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}\)
• \(\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}\)

By substituting these values into the equation, you can break down the expression step by step. These fixed values can be plugged directly into the transformed equations to aid in the simplification and help transform a complex expression into something easier to solve.

expression simplification

Simplifying trigonometric expressions often involves breaking down the expression into smaller, more manageable parts. After applying the angle addition formulas and substituting known sine and cosine values, you can start combining like terms and simplifying further.

For the example in question:

• You first substitute the angle addition transformations into the primary expression.
• Then substitute the sine and cosine values.
• Finally, group terms together and reduce to form a simpler expression. This involves distributing and combining like terms, offering a more concise representation.

The goal of simplification is to not only make the expression manageable but to also make further calculations easier, such as determining its range or using it in other equations.

range of trigonometric functions

Understanding the range of sine and cosine values is essential, especially when determining the possible range of a more complex trigonometric expression. Since both \(\sin(\theta)\) and \(\cos(\theta)\) function between -1 and 1, expressions containing these functions will also function within a range defined by the coefficients and added terms.

For example, by considering what happens when you multiply these by coefficients and combine them:

• You determine the impact on the range.
• This helps in finding max and min values the expression can take.

In our problem, correctly analyzing the transformed and simplified expression allows us to find that the final range is approximately between -7.05 and 7.05, which matches closest to the given option of -7 to 7.