Sine, Cosine & Tangent in Terms of Standard Position (College Board AP® Precalculus): Study Guide

Sine, cosine & tangent in terms of standard position

How are sine, cosine, and tangent connected to right triangle trigonometry?

• From previous courses, you may have learned sine, cosine, and tangent as ratios of side lengths in a right triangle (SOHCAHTOA)

  • $\sin \theta =\frac{\text{opposite}}{\text{hypotenuse}}$, $\cos \theta =\frac{\text{adjacent}}{\text{hypotenuse}}$, $\tan \theta =\frac{\text{opposite}}{\text{adjacent}}$

• In this course, these definitions are extended

  • using angles in standard position

  • and circles centered at the origin

• This more general approach allows sine, cosine, and tangent to be defined for any angle

  • not just acute angles in a right triangle

• However the two approaches are fully consistent

  • The right triangle definitions are a special case of the standard position definitions

What is the sine of an angle in standard position?

• Given an angle $\mathit{\theta }$ in standard position and a circle centered at the origin

  • the terminal ray of the angle

  • intersects the circle at a point $\mathit{P}$

• The sine of the angle is the ratio of

  • the vertical displacement of $P$ from the $x$-axis

  • to the distance between the origin and $P$:

$\sin \theta =\frac{\text{vertical displacement of }P\text{ from the }x\text{-axis}}{\text{distance from the origin to }P}$

• For a circle of radius $\mathit{r}$, this means

  • $\sin \theta =\frac{y}{r}$

    • where $y$ is the $y$-coordinate of point $P$

    • and $r$ is the radius of the circle

• For a unit circle (where $r=1$), this simplifies to:

  • $\sin \theta =y$

    • i.e. the sine of the angle is simply the $y$-coordinate of the point where the terminal ray meets the unit circle

What is the cosine of an angle in standard position?

• For the same angle $\mathit{\theta }$ in standard position, the cosine of the angle is the ratio of

  • the horizontal displacement of $P$ from the $y$-axis

  • to the distance between the origin and $P$:

$\cos \theta =\frac{\text{horizontal displacement of }P\text{ from the }y\text{-axis}}{\text{distance from the origin to }P}$

• For a circle of radius $\mathit{r}$

  • $\cos \theta =\frac{x}{r}$

    • where $x$ is the $x$-coordinate of point $P$

    • and $r$ is the radius of the circle

• For a unit circle (where $r=1$)

  • $\cos \theta =x$

    • i.e. the cosine of the angle is simply the $x$-coordinate of the point where the terminal ray meets the unit circle

What is the tangent of an angle in standard position?

• The tangent of an angle $\theta$ in standard position is the slope of the terminal ray

  • when the slope exists

• Because the slope of a line through the origin is the ratio of the $\mathbf{y}$-coordinate to the $\mathit{x}$-coordinate for any point on the line

  • $\tan \theta =\frac{y}{x}$

    • where $(x,y)$ is any point on the terminal ray (other than the origin)

• This is equivalent to:

  • $\tan \theta =\frac{\sin \theta }{\cos \theta }$

    • provided that $\mathbf{cos}\mathit{\theta }\mathbf{\ne }\mathbf{0}$

• The tangent is undefined when the terminal ray is vertical (i.e. when $\cos \theta =0$)

  • This occurs at $\theta =\frac{\pi }{2}+k\pi$ for integer values of $k$

    • I.e. when $\theta$ is equal to $...,-\frac{5\pi }{2},-\frac{3\pi }{2},-\frac{\pi }{2},\frac{\pi }{2},\frac{3\pi }{2},\frac{5\pi }{2},...$

How do the signs of sine, cosine, and tangent depend on the quadrant?

• The signs of the $x$- and $y$-coordinates of point $P$ change depending on which quadrant the terminal ray lies in

  • This means the signs of sine, cosine, and tangent also vary by quadrant

• A common mnemonic for remembering which trig ratios are positive in each quadrant is "All Students Take Calculus"

  • Quadrant I: All are positive

  • Quadrant II: Sine is positive

  • Quadrant III: Tangent is positive

  • Quadrant IV: Cosine is positive

How does this relate to right triangle trigonometry?

• When the terminal ray is in the first quadrant a right triangle can be formed by drawing a vertical line from $P$ down to the $x$-axis

  • The horizontal leg has length $x$

  • the vertical leg has length $y$

  • and the hypotenuse has length $r$

• The standard position definitions produce the same ratios as SOH CAH TOA

• The advantage of the standard position approach is that it works for any angle

  • including angles in the second, third, and fourth quadrants

  • as well as angles greater than $\mathbf{360}\mathbf{°}$ or negative angles

• The signs of the coordinates automatically account for the signs of sine, cosine and tangent