Sine, cosine, and tangent are ratios that tie an angle to the sides of a right triangle. Sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent, the pattern behind the mnemonic SOH-CAH-TOA. The same three ratios extend to any angle through the unit circle.

Key Takeaways

In DEG mode sin(30) → 0.5, cos(60) → 0.5, and tan(45) → 1.

sin(90) → 1, because at a quarter turn the opposite side equals the hypotenuse.

The site defaults to DEG; in RAD mode sin(30) returns about -0.988, because 30 means radians there.

asin(0.5) → 30 and atan(1) → 45 recover the angles from the ratios.

SOH-CAH-TOA and the right triangle

In a right triangle, the side opposite the right angle is the hypotenuse, and the two remaining sides are named relative to a chosen angle. The side facing the angle is the opposite side; the side that touches the angle and is not the hypotenuse is the adjacent side. The three ratios fall straight out of those names:

SOH: sine = opposite ÷ hypotenuse.
CAH: cosine = adjacent ÷ hypotenuse.
TOA: tangent = opposite ÷ adjacent.

Worked example 1: a 30° triangle with a hypotenuse of 10. Sine gives the opposite side: opposite = 10 × sin(30°) = 10 × 0.5 = 5. Cosine gives the adjacent side: adjacent = 10 × cos(30°) ≈ 10 × 0.8660254 = 8.660254. With the calculator in DEG, 10sin(30) → = 5 and 10cos(30) → = 8.660254.... Tangent ties the two sides together: tan(30°) = 5 ÷ 8.660254 ≈ 0.5774.

The unit circle extends the functions

The right-triangle picture only works for angles up to 90°, because beyond that there is no right triangle to draw. The unit circle removes that limit. Draw a circle of radius 1 around the origin and pick a point on it; the x-coordinate is the cosine of the angle from the positive x-axis, and the y-coordinate is the sine. Tangent is still sin ÷ cos. The trigonometry reference collects the identities and the value tables this circle implies.

Exact values worth knowing

A small set of angles comes up constantly, and their values are exact:

sin(0°) = 0, sin(30°) = 1/2, sin(45°) = √2/2, sin(60°) = √3/2, sin(90°) = 1.
cos(0°) = 1, cos(30°) = √3/2, cos(45°) = √2/2, cos(60°) = 1/2, cos(90°) = 0.
tan(45°) = 1, tan(30°) ≈ 0.5774, and tan(90°) is undefined because it divides by cos(90°) = 0.

Worked example 2: the 45-45-90 triangle. An isosceles right triangle has two equal legs, so opposite and adjacent are equal and tan(45°) = 1. Its hypotenuse is √2 times a leg, which is why sin(45°) = cos(45°) = 1/√2 ≈ 0.7071.

Inverse trigonometric functions

The inverse functions answer the reverse question: given a ratio, which angle produced it. asin is the inverse of sine, acos of cosine, and atan of tangent. The answer comes back in whatever angle mode the calculator is set to.

Worked example 3: recovering an angle. asin(0.5) in DEG → = 30, and atan(1) → = 45. Check the forward direction to confirm: sin(30) → = 0.5 and tan(45) → = 1. Round trips like this are a quick way to verify which mode the toggle is in.

Degrees versus radians

Degrees split a full turn into 360 parts, convenient for everyday angles. Radians measure an angle by the arc it cuts on the unit circle, so a full turn is 2π. The site defaults to DEG and you switch with the DEG/RAD toggle. The choice changes how every trig function reads the number you type: in RAD mode sin(30) returns the sine of 30 radians, about -0.988, not 0.5. The degrees vs radians guide explains both units in full, and the blog posts on degrees vs radians and pi explained cover why π/2 radians is exactly a right angle.

Where trigonometry shows up

Surveyors find building heights from one known distance and one tangent or sine. Civil engineers express road grades and slopes as tangents. Sound, light, and radio travel as sine waves. Each of those applications boils down to an angle, a side, and a trig function you can type. The calculator function reference lists every trig key and how the DEG/RAD toggle affects it.

Frequently asked questions

Should I leave the calculator in DEG or RAD?

For everyday angles measured in degrees, leave it in DEG, which is the default. Switch to RAD when you work with arc length, the unit circle, or calculus, where the formulas assume radian measure.

Why is tan(90°) undefined?

Tangent is sin ÷ cos. Since cos(90°) = 0, the division has no finite value, and the tangent curve shoots upward without ever landing on a number at 90°.

What does SOH-CAH-TOA stand for?

Sine is Opposite over Hypotenuse, Cosine is Adjacent over Hypotenuse, Tangent is Opposite over Adjacent. The first letter of each word spells the mnemonic.

Why does my calculator show sin(30) = -0.988 instead of 0.5?

Because it is in RAD mode. In radians, the input 30 means 30 radians, not 30 degrees. Flip the toggle to DEG and sin(30) → = 0.5.