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Questions and Answers
What is the reciprocal identity for cosine?
What is the reciprocal identity for cosine?
- $\cos(\theta) = \frac{1}{\tan(\theta)}$
- $\cos(\theta) = \frac{1}{\csc(\theta)}$
- $\cos(\theta) = \frac{1}{\sin(\theta)}$
- $\cos(\theta) = \frac{1}{\sec(\theta)}$ (correct)
Which trigonometric function's definition involves the ratio of the side opposite an angle to the hypotenuse in a right triangle?
Which trigonometric function's definition involves the ratio of the side opposite an angle to the hypotenuse in a right triangle?
- Cosecant (csc)
- Tangent (tan)
- Sine (sin) (correct)
- Secant (sec)
What is the reciprocal identity for sine?
What is the reciprocal identity for sine?
- $\sin(\theta) = \frac{1}{\sec(\theta)}$
- $\sin(\theta) = \frac{1}{\tan(\theta)}$
- $\sin(\theta) = \frac{1}{\csc(\theta)}$ (correct)
- $\sin(\theta) = \frac{1}{\cos(\theta)}$
What is the tangent of an angle in relation to the side lengths of a right triangle?
What is the tangent of an angle in relation to the side lengths of a right triangle?
Which trigonometric function's definition involves the ratio of the side adjacent to an angle to the hypotenuse in a right triangle?
Which trigonometric function's definition involves the ratio of the side adjacent to an angle to the hypotenuse in a right triangle?
Which identity can be used to find the value of $\cos(120^{ ext{o}})$?
Which identity can be used to find the value of $\cos(120^{ ext{o}})$?
Which identity is helpful in simplifying the expression $\cos^2(x) - \sin^2(x)$?
Which identity is helpful in simplifying the expression $\cos^2(x) - \sin^2(x)$?
Which identity is used to find $\tan(75^{ ext{o}})$ if $\tan(45^{ ext{o}}) = 1$?
Which identity is used to find $\tan(75^{ ext{o}})$ if $\tan(45^{ ext{o}}) = 1$?
Which identity is essential for deriving the double-angle formula for $\tan$?
Which identity is essential for deriving the double-angle formula for $\tan$?
Which identity can be used to simplify the expression $2[\sin(x) + \sin(y)]$?
Which identity can be used to simplify the expression $2[\sin(x) + \sin(y)]$?
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Study Notes
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
- 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.
- 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.
- 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.
- Cosecant (csc) of an angle is the reciprocal of the sine of an angle.
- Secant (sec) of an angle is the reciprocal of the cosine of an angle.
- 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
- ( \sin(\theta) = \frac{1}{\csc(\theta)} )
- ( \cos(\theta) = \frac{1}{\sec(\theta)} )
- ( \tan(\theta) = \frac{1}{\cot(\theta)} )
Pythagorean Identities
- ( \sin^2(\theta) + \cos^2(\theta) = 1 )
- ( 1 + \tan^2(\theta) = \sec^2(\theta) )
- ( 1 + \cot^2(\theta) = \csc^2(\theta) )
Cofunction Identities
- ( \sin(\frac{\pi}{2} - \theta) = \cos(\theta) )
- ( \cos(\frac{\pi}{2} - \theta) = \sin(\theta) )
Double-angle and Half-angle Identities
- ( \sin(2\theta) = 2\sin(\theta)\cos(\theta) )
- ( \cos(2\theta) = 2\cos^2(\theta) - 1 = 1 - 2\sin^2(\theta) )
- ( \tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)} )
- ( \sin(\theta) = \pm\sqrt{\frac{1 - \cos(2\theta)}{2}} )
- ( \cos(\theta) = \pm\sqrt{\frac{1 + \cos(2\theta)}{2}} )
Sum-to-Product and Product-to-Sum Identities
- ( \sin(\alpha) + \sin(\beta) = 2\sin(\frac{\alpha + \beta}{2})\cos(\frac{\alpha - \beta}{2}) )
- ( \sin(\alpha) - \sin(\beta) = 2\cos(\frac{\alpha + \beta}{2})\sin(\frac{\alpha - \beta}{2}) )
- ( \cos(\alpha) + \cos(\beta) = 2\cos(\frac{\alpha + \beta}{2})\cos(\frac{\alpha - \beta}{2}) )
- ( \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
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