삼각함수의 뜻과 관계

For the following exercises, use reference angles to evaluate the given expression.

Which trigonometric functions are odd?

We know $g(x)=\cos \,x$ is an even function, and $f(x)=\sin \,x$ and $h(x)=\tan \,x$ are odd functions. What about $G(x)={\cos }^{2}x,F(x)={\sin }^{2}x,$ and $H(x)={\tan }^{2}x?$ Are they even, odd, or neither? Why?

Examine the graph of $f(x)=\sec \,x$ on the interval $[-\pi ,\pi ].$ How can we tell whether the function is even or odd by only observing the graph of $f(x)=\sec \,x?$

After examining the reciprocal identity for $\sec \,t,$ explain why the function is undefined at certain points.

All of the Pythagorean identities are related. Describe how to manipulate the equations to get from ${\sin }^{2}t+{\cos }^{2}t=1$ to the other forms.

For the following exercises, use the fundamental identities to fully simplify the expression.

$\sin \,x\,\cos \,x\,\sec \,x$

For the following exercises, use the fundamental identities to fully simplify the expression.

$\sin (-x)\,\cos (-x)\,\csc (-x)$

For the following exercises, use the fundamental identities to fully simplify the expression.

$\tan \,x\,\sin \,x+\sec \,x\,{\cos }^{2}x$

For the following exercises, use the fundamental identities to fully simplify the expression.

$\csc \,x+\cos \,x\,\cot (-x)$

For the following exercises, use the fundamental identities to fully simplify the expression.

$\frac{\cot \,t+\tan \,t}{\sec (-t)}$

For the following exercises, use the fundamental identities to fully simplify the expression.

$3\,{\sin }^{3}\,t\,\csc \,t+{\cos }^{2}\,t+2\,\cos (-t)\cos \,t$

For the following exercises, use the fundamental identities to fully simplify the expression.

$-\tan (-x)\cot (-x)$

For the following exercises, use the fundamental identities to fully simplify the expression.

$\frac{-\sin (-x)\cos \,x\,\sec \,x\,\csc \,x\,\tan \,x}{\cot \,x}$

For the following exercises, use the fundamental identities to fully simplify the expression.

$\frac{1+{\tan }^{2}\theta }{{\csc }^{2}\theta }+{\sin }^{2}\theta +\frac{1}{{\sec }^{2}\theta }$

For the following exercises, use the fundamental identities to fully simplify the expression.

$\left(\frac{\tan \,x}{{\csc }^{2}x}+\frac{\tan \,x}{{\sec }^{2}x}\right)\left(\frac{1+\tan \,x}{1+\cot \,x}\right)-\frac{1}{{\cos }^{2}x}$

For the following exercises, use the fundamental identities to fully simplify the expression.

$\frac{1-{\cos }^{2}\,x}{{\tan }^{2}\,x}+2\,{\sin }^{2}\,x$

For the following exercises, simplify the first trigonometric expression by writing the simplified form in terms of the second expression.

$\frac{\tan \,x+\cot \,x}{\csc \,x};\,\cos \,x$

For the following exercises, simplify the first trigonometric expression by writing the simplified form in terms of the second expression.

$\frac{\sec \,x+\csc \,x}{1+\tan \,x};\,\sin \,x$

For the following exercises, simplify the first trigonometric expression by writing the simplified form in terms of the second expression.

$\frac{\cos \,x}{1+\sin \,x}+\tan \,x;\,\cos \,x$

For the following exercises, simplify the first trigonometric expression by writing the simplified form in terms of the second expression.

$\frac{1}{\sin \,x\cos \,x}-\cot \,x;\,\cot \,x$