Trigonometric Identities Quiz

Trigonometric Identities

Trigonometry is a branch of mathematics that deals with the relationships between angles and the lengths and directions of lines in a plane or a space. At the heart of trigonometry are the six fundamental trigonometric functions: sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot). These functions are interconnected through a group of relationships known as trigonometric identities.

Definitions

1. Sine (sin) of an angle is the ratio of the length of the side opposite an angle to the length of the hypotenuse in a right triangle.
2. Cosine (cos) of an angle is the ratio of the length of the side adjacent to an angle to the length of the hypotenuse in a right triangle.
3. Tangent (tan) of an angle is the ratio of the length of the side opposite an angle to the length of the side adjacent to an angle in a right triangle.
4. Cosecant (csc) of an angle is the reciprocal of the sine of an angle.
5. Secant (sec) of an angle is the reciprocal of the cosine of an angle.
6. Cotangent (cot) of an angle is the reciprocal of the tangent of an angle.

Trigonometric Identities

Trigonometric identities are equations or statements that are true for all angles within a specified domain. There are several types of trigonometric identities, but we'll focus on the most important ones:

Reciprocal Identities

1. ( \sin(\theta) = \frac{1}{\csc(\theta)} )
2. ( \cos(\theta) = \frac{1}{\sec(\theta)} )
3. ( \tan(\theta) = \frac{1}{\cot(\theta)} )

Pythagorean Identities

1. ( \sin^2(\theta) + \cos^2(\theta) = 1 )
2. ( 1 + \tan^2(\theta) = \sec^2(\theta) )
3. ( 1 + \cot^2(\theta) = \csc^2(\theta) )

Cofunction Identities

1. ( \sin(\frac{\pi}{2} - \theta) = \cos(\theta) )
2. ( \cos(\frac{\pi}{2} - \theta) = \sin(\theta) )

Double-angle and Half-angle Identities

1. ( \sin(2\theta) = 2\sin(\theta)\cos(\theta) )
2. ( \cos(2\theta) = 2\cos^2(\theta) - 1 = 1 - 2\sin^2(\theta) )
3. ( \tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)} )
4. ( \sin(\theta) = \pm\sqrt{\frac{1 - \cos(2\theta)}{2}} )
5. ( \cos(\theta) = \pm\sqrt{\frac{1 + \cos(2\theta)}{2}} )

Sum-to-Product and Product-to-Sum Identities

1. ( \sin(\alpha) + \sin(\beta) = 2\sin(\frac{\alpha + \beta}{2})\cos(\frac{\alpha - \beta}{2}) )
2. ( \sin(\alpha) - \sin(\beta) = 2\cos(\frac{\alpha + \beta}{2})\sin(\frac{\alpha - \beta}{2}) )
3. ( \cos(\alpha) + \cos(\beta) = 2\cos(\frac{\alpha + \beta}{2})\cos(\frac{\alpha - \beta}{2}) )
4. ( \cos(\alpha) - \cos(\beta) = -2\sin(\frac{\alpha + \beta}{2})\sin(\frac{\alpha - \beta}{2}) )

These identities help us to simplify and manipulate trigonometric expressions, and they form the foundation of a wide range of trigonometric theorems and problems. Wikipedia contributors. (2023, January 23). Trigonometric functions. Wikipedia. [Online] Available at: https://en.wikipedia.org/wiki/Trigonometric_functions Wikipedia contributors. (2023, January 23). Trigonometric identities. Wikipedia. [Online] Available at: https://en.wikipedia.org/wiki/Trigonometric_identities