Question

2 exploration: writing other trigonometric identities work with a partner. the trigonometric identity you derived in exploration 1 is called a pythagorean identity. there are two other pythagorean identities. to derive them, recall the four relationships: \\(\tan \theta = \frac{\sin \theta}{\cos \theta}\\) \\(\cot \theta = \frac{\cos \theta}{\sin \theta}\\) \\(\sec \theta = \frac{1}{\cos \theta}\\) \\(\csc \theta = \frac{1}{\sin \theta}\\) a. divide each side of the pythagorean identity you derived in exploration 1 by \\(\cos^2 \theta\\) and simplify. what do you observe? b. divide each side of the pythagorean identity you derived in exploration 1 by \\(\sin^2 \theta\\) and simplify. what do you observe? communicate your answer 3. how can you verify a trigonometric identity? 4. is \\(\sin \theta = \cos \theta\\) a trigonometric identity? explain your reasoning. 5. give some examples of trigonometric identities that are different than those in explorations 1 and 2.

Explanation:

Step1: Understand trigonometric identity verification

To verify a trigonometric identity, we can start from one side of the equation and use known trigonometric identities (such as reciprocal identities \( \tan\theta=\frac{\sin\theta}{\cos\theta},\cot\theta = \frac{\cos\theta}{\sin\theta},\sec\theta=\frac{1}{\cos\theta},\csc\theta=\frac{1}{\sin\theta}\) and Pythagorean identities \( \sin^{2}\theta+\cos^{2}\theta = 1,1+\tan^{2}\theta=\sec^{2}\theta,1+\cot^{2}\theta=\csc^{2}\theta\)) to transform it into the other side. We can also simplify both sides of the equation separately and show that they are equal.

Step2: Analyze \( \sin\theta=\cos\theta\)

A trigonometric identity is an equation that is true for all values of the variable for which both sides are defined. For \(y = \sin\theta\) and \(y=\cos\theta\), when \(\theta = \frac{\pi}{4}+ 2k\pi,k\in\mathbb{Z}\), \(\sin\theta=\cos\theta=\frac{\sqrt{2}}{2}\); when \(\theta=\frac{5\pi}{4}+2k\pi,k\in\mathbb{Z}\), \(\sin\theta=\cos\theta=-\frac{\sqrt{2}}{2}\). But for example, when \(\theta = 0\), \(\sin(0)=0\) and \(\cos(0) = 1\), so \(\sin\theta=\cos\theta\) is not True for all \(\theta\) for which \(\sin\theta\) and \(\cos\theta\) are defined.

Step3: Provide examples of other trigonometric identities

Double - angle formulas: \(\sin(2\theta)=2\sin\theta\cos\theta\), \(\cos(2\theta)=\cos^{2}\theta-\sin^{2}\theta = 2\cos^{2}\theta - 1=1 - 2\sin^{2}\theta\), \(\tan(2\theta)=\frac{2\tan\theta}{1-\tan^{2}\theta}\). Sum - of - angles formulas: \(\sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta\), \(\cos(\alpha + \beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta\), \(\tan(\alpha+\beta)=\frac{\tan\alpha+\tan\beta}{1 - \tan\alpha\tan\beta}\)

Answer:

1. We can verify a trigonometric identity by using known trigonometric identities (reciprocal, Pythagorean, etc.) to transform one side of the equation to the other side or simplify both sides separately to show their equality.
2. No, \(\sin\theta=\cos\theta\) is not a trigonometric identity because it is not True for all values of \(\theta\) (e.g., \(\theta = 0\), \(\sin(0)=0

eq\cos(0) = 1\)).

1. Examples: Double - angle formulas (\(\sin(2\theta)=2\sin\theta\cos\theta\), \(\cos(2\theta)=2\cos^{2}\theta - 1\) etc.) and sum - of - angles formulas (\(\sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta\) etc.)