91Ó°ÊÓ

The six trigonometric functions are: 1. Sine (\(\sin x\)) 2. Cosine (\(\cos x\)) 3. Tangent (\(\tan x\)) 4. Cotangent (\(\cot x\)) 5. Secant (\(\sec x\)) 6. Cosecant (\(\csc x\))

To check if the sine function can derive all the other trigonometric functions, we need to look at some trigonometric identities and relationships. 1. \(\sin^2{x} + \cos^2{x} = 1\) The equation above shows a relation between sine and cosine. Solving for cosine, we get: \(\cos x = \pm\sqrt{1 - \sin^2 x}\) 2. \(\tan x = \frac{\sin x}{\cos x}\) By substituting the expression for cosine from the relation in step 1, we can express tangent in terms of sine: \(\tan x = \frac{\sin x}{\pm\sqrt{1 - \sin^2 x}}\) 3. \(\cot x = \frac{\cos x}{\sin x}\) Using the cosine expression from step 1, we can express cotangent in terms of sine: \(\cot x = \frac{\pm\sqrt{1 - \sin^2 x}}{\sin x}\) 4. \(\sec x = \frac{1}{\cos x}\) Using the cosine expression from step 1, we can express secant in terms of sine: \(\sec x = \frac{1}{\pm\sqrt{1 - \sin^2 x}}\) 5. \(\csc x = \frac{1}{\sin x}\) Cosecant is already expressed in terms of the sine function. All six trigonometric functions can be expressed in terms of sine using the trigonometric relationships and identities we've shown above. Therefore, your friend is correct that all six trigonometric functions can be obtained from the sine function.