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Math101learn.math101.caExact 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.
The 45–45–90 triangle
Cut a square along its diagonal. If each leg is $1$, the Pythagorean theorem gives hypotenuse $\sqrt2$. Therefore, for $45^\circ$,
The sine and cosine match because the opposite and adjacent legs are equal.
The 30–60–90 triangle
Bisect an equilateral triangle of side length $2$. Each half has hypotenuse $2$, short leg $1$, and long leg $\sqrt3$.
For $30^\circ$:
For $60^\circ$:
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: $\cos(-\theta)=\cos\theta$.
Quadrantal values
The unit-circle coordinates give values at axes:
| Angle | Point | $\cos\theta$ | $\sin\theta$ | $\tan\theta$ |
|---|---|---|---|---|
| $0^\circ$ | $(1,0)$ | $1$ | $0$ | $0$ |
| $90^\circ$ | $(0,1)$ | $0$ | $1$ | undefined |
| $180^\circ$ | $(-1,0)$ | $-1$ | $0$ | $0$ |
| $270^\circ$ | $(0,-1)$ | $0$ | $-1$ | undefined |
Tangent is undefined when cosine, its denominator, equals zero.
Exact values in radians
The process is identical in radians. For example,
so its reference angle is $\pi/6$ and
Recognizing benchmark radian angles avoids unnecessary conversion every time.
Deriving instead of memorizing
If a value is forgotten, reconstruct the appropriate special triangle or use the unit-circle coordinate. This is more reliable than memorizing a large disconnected chart.
The identity
can derive tangent once sine and cosine are known.
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?
Related topics
Explore the idea
Triangle and angle explorer
Change one quantity at a time and connect what moves to Exact Trigonometric Values.
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
What is the exact value of sin 150°?
- Reference angle = 180° − 150° = 30°.
- sin 30° = 1/2.
- Sine is positive in Quadrant II, so sin 150° = 1/2.
End of lesson
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