You memorized sin 30° = 1/2, then panicked because you could not tell whether cos 150° is positive or negative. The two classic failures of trigonometry are forgetting values and attaching the wrong sign.
The fix in one sentence: every special-angle value — 0°, 30°, 45°, 60°, 90° — can be derived from just two right triangles. This article covers the derivations, the degree-to-radian conversion, and obtuse angles, then verifies everything with a free browser tool.
What sine, cosine and tangent actually measure
In a right triangle, each trig ratio compares two sides relative to an angle θ:
| Ratio | Definition |
|---|---|
| sin θ | opposite ÷ hypotenuse |
| cos θ | adjacent ÷ hypotenuse |
| tan θ | opposite ÷ adjacent |
With the definition in hand, values become side-counting — two triangles cover all of them.
30° and 60°: the 1 : √3 : 2 triangle
The values for 30° and 60° come from a right triangle with sides in the ratio 1 : √3 : 2 — half of an equilateral triangle.
The ratio comes from cutting an equilateral triangle of side 2 through its apex: the base splits into 1 + 1, and the height h follows from 1² + h² = 2², giving h = √3.
- sin 30° = 1/2
- cos 30° = √3/2
- tan 30° = 1/√3 = √3/3
- Swapping the roles of the sides gives sin 60° = √3/2, cos 60° = 1/2, tan 60° = √3
45°: the 1 : 1 : √2 triangle
For 45°, use an isosceles right triangle with sides 1 : 1 : √2 (a square cut along its diagonal):
- sin 45° = 1/√2 = √2/2
- cos 45° = √2/2
- tan 45° = 1/1 = 1
Opposite and adjacent sides are equal — hence sin 45° = cos 45°. The axis angles follow by letting the triangle collapse: sin 0° = 0, cos 0° = 1, sin 90° = 1, cos 90° = 0, and tan 90° = 1 ÷ 0 is undefined.
Converting degrees to radians
A radian measures the angle by arc length on a unit circle, anchored by the relation 180° = π radians. To convert, multiply degrees by π/180:
| Degrees | Radians |
|---|---|
| 30° | π/6 |
| 45° | π/4 |
| 60° | π/3 |
| 90° | π/2 |
| 120° | 2π/3 |
| 150° | 5π/6 |
| 180° | π |
For example, 30° × π/180 = π/6; to go back, multiply by 180/π. Strange calculator output usually means a unit-mode mismatch.
Radians are the standard in mathematics and physics: arc length is simply s = rθ, while degrees drag a conversion factor into every formula.
Obtuse angles: quadrants and the reference angle
For angles past 90°, read values in two steps: take the value from the reference angle, then apply the sign of the quadrant. In quadrant II (90°–180°) only sin is positive; cos and tan are negative. For cos 150°:
- 150° = 180° − 30°, so the reference angle is 30°.
- Quadrant II makes cos negative.
- Therefore cos 150° = −cos 30° = −√3/2.
Similarly sin 150° = sin 30° = 1/2 and tan 150° = −tan 30° = −√3/3. In quadrant III (180°–270°) tan is the only positive one: at 225° the reference angle is 45°, and tan 225° = (−√2/2) ÷ (−√2/2) = 1 because dividing two negatives cancels the sign.
Three common mistakes: leaving the calculator in the wrong angle mode (if sin π does not come out 0, suspect the mode); trying to evaluate tan at 90° or 270°, where cos = 0 makes it undefined; and dropping the negative sign in quadrant II. The habits "confirm the unit, then confirm the sign" prevent all three.
Verify values with the tool
For checking hand calculations, Tools Hub’s Trigonometry Calculator shows sin, cos and tan of any angle at once, outputs exact values for special angles, accepts degrees and radians (including expressions like pi/6), and displays the quadrant signs with the reference-angle working.
How to use it (3 steps)
Pick the unit
Choose between the Degrees and Radians tabs to match your problem.
Enter the angle
Type a number (150) or a pi expression (5pi/6). Preset buttons cover 30°, 45°, 60° and 90°.
Read values and reasoning
Besides sin, cos and tan you get the conversion to the other unit, the quadrant signs, and the reference-angle explanation — enough to check your signs, not just your values.
The tool featured in this article
Trigonometry Calculator
sin, cos and tan together, exact values for special angles, degree and radian support with quadrant and reference-angle display — free and browser-based.
To reproduce the article’s example, set the tool to Degrees, enter 150, and read sin = 1/2, cos = −√3/2, tan = −√3/3 along with the reference-angle note. Switching to Radians and entering pi/6 gives sin = 1/2 again. For tidying the fractions inside trig values, see the guide to working with fractions.
Summary
- Two triangles — 1 : √3 : 2 and 1 : 1 : √2 — generate every special-angle value from the side-ratio definitions
- Convert with 180° = π: multiply degrees by π/180
- Past 90°, read “value from the reference angle, sign from the quadrant” (quadrant II: only sin is positive)
- tan 90° is undefined because cos = 0; strange calculator output usually means a unit-mode mismatch
FAQ
Why derive values instead of memorizing them?
Derivations survive forgetting. All special angles reduce to two triangles, so what you memorize is the two side ratios plus the three definitions.
Why is sin 90° equal to 1?
Picture a unit circle. A point standing at 90° sits at the very top: its height (sin) is 1 and its horizontal position (cos) is 0. Reading the ratios as height, width and slope also explains the quadrant signs.
Why can’t tan 90° be computed?
tan = sin/cos, and cos 90° = 0, so the division has no meaning — the graph runs along an asymptote there.
How can I tell whether my calculator is in degrees or radians?
Compute sin 180. If you get 0, it is in degrees; if you get almost 0, it is in radians (since sin π = 0). The tool avoids the issue by asking you to pick the unit.
References
- OpenStax, Algebra and Trigonometry 2e, Section 7.1 “Angles” (degree–radian conversion, reference angles): openstax.org
- OpenStax, Algebra and Trigonometry 2e, Section 7.2 “Right Triangle Trigonometry” (trig ratios and special angles): openstax.org
