Fundamental Trigonometric Identities


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Reciprocal Identities

Recall

\(\csc\left(\theta\right)=\dfrac{1}{\sin\left(\theta\right)}\)
\(\sec\left(\theta\right)=\dfrac{1}{\cos\left(\theta\right)}\)
\(\cot\left(\theta\right)=\dfrac{1}{\tan\left(\theta\right)}\)

 

 

Quotient Identities

Identities can be manipulated to make new identities.

 

Proving Identities

Math in Action

New mathematical identities can be developed or proven by beginning with a known identity and using algebra to find an equivalent equation.

 

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Sine, Cosine, and the Unit Circle

Since \(P\left(\cos\left(\theta\right),\sin\left(\theta\right)\right)\) is a point on the unit circle \(x^2+y^2=1\), for an angle \(\theta\) in standard position, \(\cos^2\left(\theta\right)+\sin^2\left(\theta\right)=1\).

The right triangle is formed by drawing a vertical line segment from P to the x-axis. The terminal arm passing through point P makes up the third side of the triangle.

This is another fundamental identity we will be using in this lesson. It gets its name from the right triangle in the image.

Pythagorean Identity

\[\cos^2\left(\theta\right)+\sin^2\left(\theta\right)=1\]

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