Math101 · Independent Ontario learning library
Math101Exact Trigonometric Values
Special triangles, reference angles, and quadrant signs produce exact sine, cosine, and tangent values without decimals.
Exact values preserve the geometry of special angles using fractions and radicals instead of rounded calculator decimals.
Why exact form matters
A decimal such as $0.7071$ approximates $\sqrt2/2$. Exact form keeps the complete value, supports later algebra, and avoids accumulated rounding error.
The most common special acute angles are $30^\circ$, $45^\circ$, and $60^\circ$, or $\pi/6$, $\pi/4$, and $\pi/3$ radians.
Quadrant signs
Reference angles provide the magnitude; the terminal arm's quadrant provides the sign.
| Quadrant | Positive primary ratios |
|---|---|
| I | sine, cosine, tangent |
| II | sine only |
| III | tangent only |
| IV | cosine only |
Coordinate definitions explain the pattern: sine follows $y$, cosine follows $x$, and tangent follows $y/x$.
Worked example: an angle in Quadrant II
Worked example: a negative angle
Find $\cos(-45^\circ)$. The angle is coterminal with $315^\circ$, which lies in Quadrant IV and has reference angle $45^\circ$. Cosine is positive there, so
This also illustrates that cosine is an even function: $\cos(-\theta)=\cos\theta$.
Common mistakes
Giving a decimal when exact form is requested. Keep fractions and radicals.
Using the reference angle's positive value without a sign check. Apply the quadrant sign.
Swapping the long and short legs. The side opposite $60^\circ$ is longer.
Calling tangent at $90^\circ$ zero. It is undefined because cosine is zero.
Mixing degree and radian labels. $30$ and $\pi/6$ need the correct units.
Quick self-check
- What is the reference angle?
- Which special triangle or axis point applies?
- Which trig ratios are positive in the quadrant?
- Is the result exact and simplified?
- Can tangent be checked using sine divided by cosine?
