# Trigonometric functions --- ## CAUTION: Radians and Degrees When you are calculating trigonometric using a calculator or software, make sure you are using degrees and not radians --- ## Right Triangles Can use Sine, Cosine, and Tangent to determine the proportions of sides in right triangles  --- ## Circles Can use Sine, and Cosine to determine the x and y values on a circle  Note: We will start by explaining sin an cos with circles and then go back on how to use them in right triangles ---  $x=\cos(\theta)$ $y=\sin(\theta)$ Note: Finding the x and y components of a circle ---  $x^2 + y^2 = 1$ $\left( \cos(\theta) \right)^2 + \left( \sin(\theta) \right)^2=1$ ---  $x=\cos(0^\circ)=1$ $y=\sin(0^\circ)=0$ ---  One-third of the way between $0^\circ$ and $90^\circ$ ---  $y=\sin(30^\circ)=\frac{1}{2}$ $x^2 + y^2=1$ ---  $x^2=1-y^2$ $1-(\frac{1}{2})^2$ ---  $x^2=1-\frac{1}{4}$ $x^2=\frac{3}{4}$ ---  $x=\cos(30^\circ)=\sqrt{\frac{3}{4}}$ $y=\sin(30^\circ)=\frac{1}{2}$ ---  Halfway between $0^\circ$ and $90^\circ$ ---  $x=y$ $x^2+y^2=1$ ---  $2x^2=1$ $x^2=\frac{1}{2}$ ---  $x=\cos(45^\circ)=\sqrt{\frac{1}{2}}$ $y=\sin(45^\circ)=\sqrt{\frac{1}{2}}$ ---  Two-thirds of the way between $0^\circ$ and $90^\circ$ ---  $x=\cos(60^\circ)=\frac{1}{2}$ $y=\sin(60^\circ)=\sqrt{\frac{3}{4}}$ ---  $x=\cos(90^\circ)=0$ $y=\sin(90^\circ)=1$ --- ## Common Values of Sin and Cos | $\theta$ | $x=\cos(\theta)$ | $y=\sin(\theta)$ | | ------------ | -------------------- | -------------------- | | $0^\circ$ | 1 | 0 | | $30^\circ$ | $\sqrt{\frac{3}{4}}$ | $\frac{1}{2}$ | | $45^\circ$ | $\sqrt{\frac{1}{2}}$ | $\sqrt{\frac{1}{2}}$ | | $60^\circ$ | $\frac{1}{2}$ | $\sqrt{\frac{3}{4}}$ | | $90^\circ$ | 0 | 1 | --- ## Sohcahtoa Soh.. $Sine=\frac{Opposite}{Hypotenuse}$ ...cah... $Cosine=\frac{Adjacent}{Hypotenuse}$ ...toa... $Tangent=\frac{Opposite}{Adjacent}$ ---  $Sine=\frac{Opposite}{Hypotenuse}$ $Cosine=\frac{Adjacent}{Hypotenuse}$ ---  $\tan(\theta)=\frac{\sin(\theta)}{\cos(\theta)}$ $Tangent=\frac{Opposite}{Adjacent}$ --- ## Summary $x=\cos(\theta)$ $y=\sin(\theta)$ $\tan(\theta)=\frac{\sin(\theta)}{\cos(\theta)}$ $x^2 + y^2 = 1$ $\left( \cos(\theta) \right)^2 + \left( \sin(\theta) \right)^2=1$ --- ## Summary (Continued) Soh.. $Sine=\frac{Opposite}{Hypotenuse}$ ...cah... $Cosine=\frac{Adjacent}{Hypotenuse}$ ...toa... $Tangent=\frac{Opposite}{Adjacent}$ --- ## Summary (Continued) | $\theta$ | $x=\cos(\theta)$ | $y=\sin(\theta)$ | | ------------ | -------------------- | -------------------- | | $0^\circ$ | 1 | 0 | | $30^\circ$ | $\sqrt{\frac{3}{4}}$ | $\frac{1}{2}$ | | $45^\circ$ | $\sqrt{\frac{1}{2}}$ | $\sqrt{\frac{1}{2}}$ | | $60^\circ$ | $\frac{1}{2}$ | $\sqrt{\frac{3}{4}}$ | | $90^\circ$ | 0 | 1 |