Trigonometry in Terms of Algebra

Relationships between the lengths of two sides of a triangle.

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Composition of Trigonometric Functions and Their Inverses

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composite function is a set of two different trigonometric functions applied to an argument in conjunction with one another.

\begin{align*}\sin(\sin^{-1}(x)) = x \cos(\cos^{-1}(x)) = x \tan(\tan^{-1}(x)) = x\end{align*}

\begin{align*}\sin^{-1}(\sin(x)) = x \cos^{-1}(\cos(x)) = x \tan^{-1}(\tan(x)) = x\end{align*}

Using the inverse of a trigonometric functions allows an argument be solved.

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Trigonometry in Terms of Algebra

Remeber: All of the trig functions can be rewritten in terms of x.

\begin{align*}\tan \theta = \frac{x}{1}\\ \tan \theta = x hypotenuse = \sqrt{x^2+1}\\ \theta = \tan^{-1} x\end{align*}

\begin{align*}\sin (\tan^{-1}x) = \sin \theta = \frac{x}{\sqrt{x^2+1}} \csc (\tan^{-1}x) = \csc \theta = \frac{\sqrt{x^2+1}}{x}\\ \cos (\tan^{-1}x) = \cos \theta = \frac{1}{\sqrt{x^2+1}} \sec (\tan^{-1}x) = \sec \theta = \sqrt{x^2+1}\\ \tan (\tan^{-1}x) = \tan \theta = x \cot (\tan^{-1}x) = \cot \theta = \frac{1}{x}\end{align*}

Using the triangle above what type of trigonometric functions would use to solve for the triangle?

By using the Pythagoream Theorem, wou could solve for the hypotenuse of the triangle but the answer will not be a integer, instead it will be more of an expression with a variable.

ex:\begin{align*}\sqrt{x^2+1}\end{align*}

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Real Life Applications

Using trigonometricn functions and their inverses, you can solve real world word problem.

Tip: Always draw a diagram after reading the word problem so that it is easier to visualize the problem and make it easier for you to solve the problem.

EX: A tower, 28.4 feet high, must be secured with a guy wire anchored 5 feet from the base of the tower. What angle will the guy wire make with the ground?