91Ó°ÊÓ

The double-angle formula is a vital trigonometric identity. For the cosine function, it is given by: ewlineewline\text{Recall the double-angle formula for cosine:} ewlineewline\text{ \textbackslash [ \textbackslash cos 2\theta = 2\cos^2 \theta - 1 \textbackslash] }ewlineewline\text{Using this formula helps us find values for \(2\theta\) quickly by knowing the values for just \(\theta\). In the given exercise, we start with \(\textbackslash cos 2\theta = \frac{2}{3}\) and solve the equation: } ewlineewline\text{\textbackslash [ \textbackslash cos 2\theta = \frac{2}{3} \rightarrow \frac{2}{3} = 2\cos^2 \theta - 1 \rightarrow \cos^2 \theta = \frac{5}{6} \rightarrow \cos \theta = \textbackslash pm \textbackslash sqrt{\frac{5}{6}} \textbackslash] }ewlineewline\text{By solving, we see that knowing one double-angle value, we can retrieve the original angle's trigonometric values.}

The Pythagorean Identity is one of the cornerstones of trigonometry. It links sine and cosine in a powerful way: ewlineewline\text{\textbackslash[ \textbackslash sin^2 \theta + \textbackslash cos^2 \theta = 1 \textbackslash] }ewline\text {We used this identity in our exercise after finding \(\textbackslash cos^2 \theta\) . With \(\textbackslash cos^2 \theta = \textbackslash \frac{5}{6}\) , we determine \(\textbackslash sin \theta\)ewlineewline\text } ewlineewline\text{ \textbackslash [ \textbackslash sin^2 \theta = 1 - \textbackslash \frac{5}{6} } ewline \text {\textbackslash sin ^2 \theta = \textbackslash frac {1} {6} \textbackslash] } } }ewlineewline\text{Taking the square root gives \(\textbackslash sin \theta = \textbackslash pm \textbackslash sqrt{\frac{1}{6}}\).In our exercise, since \(\theta\) is in the second quadrant, \( \ sin \theta \) is positive.The Pythagorean identity is always useful to transition between sine and cosine values. This simplicity makes problem-solving much easier.}

The triple-angle identity lets us express trigonometric functions when the angle is tripled: ewlineewline\text {For cosine and sine, these identities are: }ewlineewline\text{\textbackslash[ cos 3\theta = 4 \cos ^3 \theta -3 \cos \theta \textbackslash] }ewlineewline\text{ \textbackslash[ \textbackslash sin 3\theta = 3 \sin \theta - 4 \sin ^3 \theta \textbackslash] }ewlineewline\text {In the exercise, after finding \(\textbackslash cos \theta \) as \(- \textbackslash sqrt{ \textbackslash frac{5}{6} } , we use triple-angle formula to solve for \cos 3\theta \)ewlineewline\text{ \textbackslash[ \textbackslash cos 3\theta = 4 ( \frac{-5 \textbackslash sqrt{5/6}} )^3 - 3 (\frac{-5 \ textbackslash sqrt{5/6} ) \textbackslash] }ewlineewline\text{By directly substituting, since it simplifies multi-angles.}

In trigonometry, the quadrant where the angle is located determines the sign of sine and cosine values. The second quadrant includes angles between \(90^\textbackslash circ \) and \(180^\textbackslash circ\): ewlineewline\text{In this range: }ewlineewline\text{ \theta (90^circ < \theta < 180^circ) ,cos \theta \text{is always negative.}} ewline {\textbackslash sin \theta \text{is always positive.}} ewline These signs impact both cosine and sine values. It's crucial for accurate trigonometric calculations. In our exercise, knowing \(\theta\) is in the second quadrant ensures correct signs which influence \(\textbackslash sin \theta \) and \( \cos \theta \).Their fundamental understanding helps identify any angle's correct trigonometric properties.