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Mobile App Chapter 19: Problem 1812
If \(\mathrm{K}\left[\sin 18^{\circ}+\cos 36^{\circ}\right]=5\) then \(\mathrm{K}=\) (a) \(2 \sqrt{5}\) (b) \((\sqrt{5} / 2)\) 4 (d) 5
Short Answer
Expert verified
The value of K is 4.
Step by step solution
01
Rewrite sin(18) using half-angle formula
Recall the half-angle formula for sine: \(\sin(\frac{x}{2}) = \pm\sqrt{\frac{1 - \cos x}{2}}\).
We can rewrite sin(18) using the half-angle formula.
\[\sin 18^{\circ} = \sqrt{\frac{1 - \cos 36^{\circ}}{2}}\]
02
Rewrite cos(36) using double-angle formula
Recall the double angle formula for cosine: \(\cos 2x = 2\cos^2 x - 1\).
Rewrite cos(36) using this formula:
\[\cos 36^{\circ} = \frac{1 + \cos 72^{\circ}}{2}\]
03
Plug in the rewritten expressions for sin(18) and cos(36) into the original equation
Substitute the expressions from steps 1 and 2 into the given equation:
\[K\left[\sqrt{\frac{1 - \cos 36^{\circ}}{2}} + \frac{1 + \cos 72^{\circ}}{2}\right] = 5\]
04
Further simplify the equation in cosines terms
Recall that \(\cos 2x = 1 - 2\sin^2 x\). So now, rewrite \(\cos 72^{\circ}\) in terms of sin:
\[\cos 72^{\circ} = 1 - 2\sin^2 36^{\circ}\]
Substitute this back into the expression and simplify:
\[K\left[\sqrt{\frac{1 - \cos 36^{\circ}}{2}} + \frac{1 + (1 - 2\sin^2 36^{\circ})}{2}\right] = 5\]
05
Simplify and solve for K
Now it's time to simplify and solve for K:
\[K\left[\sqrt{\frac{1 - \cos 36^{\circ}}{2}} + \frac{1 + 1 - 2\sin^2 36^{\circ}}{2}\right] = 5\]
\[K\left[\sqrt{\frac{1 - \cos 36^{\circ}}{2}} + \frac{2 - 2\sin^2 36^{\circ}}{2}\right] = 5\]
Notice that: \(\cos 36^{\circ} = \frac{1}{2} + \frac{\sqrt{5}}{2}\) and \(sin^2 36^{\circ} = 1 - \cos^2 36^{\circ} = \frac{5 - \sqrt{5}}{4}\)
Plug in these values into the last equation:
\[K\left[\sqrt{\frac{1 - \left(\frac{1}{2} + \frac{\sqrt{5}}{2}\right)}{2}} + \frac{2 - 2\left(\frac{5 - \sqrt{5}}{4}\right)}{2}\right] = 5\]
Solve for K:
\[K\left[\sqrt{\frac{\frac{1}{2} - \frac{\sqrt{5}}{2}}{2}} + \frac{\frac{3\sqrt{5}}{2}}{2}\right] = 5\]
Multiply K through to obtain:
\(K\sqrt{\frac{\frac{1}{2} - \frac{\sqrt{5}}{2}}{2}} + K\frac{3\sqrt{5}}{4} = 5\)
Now plug in each option (a, b, c, and d) for K, and see which one makes the equation true.
Option (c) K=4:
\[4\sqrt{\frac{\frac{1}{2} - \frac{\sqrt{5}}{2}}{2}} + 4\frac{3\sqrt{5}}{4} = 5\]
Simplify:
\[4\sqrt{\frac{\frac{1}{2} - \frac{\sqrt{5}}{2}}{2}} + 3\sqrt{5} = 5\]
This makes the expression true, so the answer is:
K = 4
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding Half-Angle Formulas
The half-angle formulas are invaluable tools when dealing with trigonometric equations, especially when you encounter an angle for which the trigonometric value isn't straightforwardly known. A half-angle formula allows us to express a trigonometric function of half an angle in terms of the function of the full angle. This is particularly useful because it can transform an equation with angles that are not standard into one that involves only standard angles, for which values are well-known or easier to calculate.
For the sine function, the half-angle formula is stated as: \[\begin{equation} \sin(\frac{x}{2}) = \pm\sqrt{\frac{1 - \cos(x)}{2}} \end{equation}\]In our exercise, this was applied to \(\sin(18^\circ)\), which is not a commonly known trigonometric value, using \(\cos(36^\circ)\) which is.
For the sine function, the half-angle formula is stated as: \[\begin{equation} \sin(\frac{x}{2}) = \pm\sqrt{\frac{1 - \cos(x)}{2}} \end{equation}\]In our exercise, this was applied to \(\sin(18^\circ)\), which is not a commonly known trigonometric value, using \(\cos(36^\circ)\) which is.
Applying Double Angle Formulas
Double angle formulas, which are another set of trigonometric tools, make it possible to express trigonometric functions of twice an angle in terms of the single angle. These formulas are immensely useful when dealing with trigonometric identities involving double angles or when reducing the degree of trigonometric polynomials.
The formula for the cosine of a double angle is: \[\begin{equation} \cos(2x) = 2\cos^2(x) - 1 \end{equation}\]By utilizing this formula, we transformed \(\cos(36^\circ)\) to be written in terms of \(\cos(72^\circ)\). This is a clever manipulation since it allowed us to deal with \(\cos(36^\circ)\) more simply in the subsequent steps.
The formula for the cosine of a double angle is: \[\begin{equation} \cos(2x) = 2\cos^2(x) - 1 \end{equation}\]By utilizing this formula, we transformed \(\cos(36^\circ)\) to be written in terms of \(\cos(72^\circ)\). This is a clever manipulation since it allowed us to deal with \(\cos(36^\circ)\) more simply in the subsequent steps.
Using Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that are true for every value of the occurring variables. They serve as the foundation for simplifying trigonometric expressions and solving trigonometric equations. In solving our problem, we employed the identity: \[\begin{equation} \cos(2x) = 1 - 2\sin^2(x) \end{equation}\]This is particularly useful in expressing cosine in terms of sine or vice versa, aiding in the simplification of expressions involving multiple trigonometric functions. These identities are a key component of the toolkit that enables the solution of complex trigonometric equations.
Mastering Algebraic Manipulation
Algebraic manipulation involves the skillful rearrangement and simplification of mathematical expressions and equations. It is fundamental to solving a wide range of mathematical problems, not least trigonometric equations. In our exercise, algebraic manipulation was essential in the final steps to simplify the equation and solve for K. We were able to isolate K and then methodically test each answer choice to find the correct solution.
To improve students' understanding, we employed a step-by-step simplification process, allowing them to follow the logical progression from the given equation to the final answer. Clear, concise, and sequential steps are crucial in teaching the algebraic manipulation needed to solve trigonometric equations like the one in this exercise.
To improve students' understanding, we employed a step-by-step simplification process, allowing them to follow the logical progression from the given equation to the final answer. Clear, concise, and sequential steps are crucial in teaching the algebraic manipulation needed to solve trigonometric equations like the one in this exercise.
