Trigonometry Calculator NZ

This calculator solves triangles and core trigonometric functions for NCEA Level 1 to 3 maths, covering right-angled triangles, any general triangle, and standalone sine, cosine and tangent calculations. For a right triangle, enter any two known values, two sides, or one side and one angle, in degrees or radians, and it works out the remaining sides, angles, area and perimeter, since the third angle is always 90 degrees. For any other triangle, enter three known values including at least one side, and it automatically applies the sine rule or cosine rule to return all three sides, all three angles, the area and the perimeter, showing which rule it used. A third tab lets you type a single angle to get sin, cos and tan plus their reciprocals, cosec, sec and cot, alongside an interactive unit circle that plots the angle and its sine and cosine components as you type. A fourth tab lists exact values for common angles from 0 to 360 degrees, useful for memorising the values NCEA expects you to know without a calculator. Each solved triangle also appears as a labelled diagram so you can check the shape matches what you expected. Use it to check homework, work through sine rule and cosine rule word problems such as bearings and surveying questions, or explore how angle and side values relate to each other.

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Trigonometry for NCEA students

Trigonometry is the study of relationships between the sides and angles of triangles. It is a core topic in NCEA Mathematics from Level 1 onwards. The three primary functions, sine (sin), cosine (cos), and tangent (tan), each describe a ratio between two sides of a right-angled triangle relative to one of the acute angles.

The most important memory aid for right triangles is SOH-CAH-TOA: sin = Opposite/Hypotenuse, cos = Adjacent/Hypotenuse, tan = Opposite/Adjacent. Here "opposite" means the side directly across from the angle you are working with, "adjacent" means the side next to it (that is not the hypotenuse), and the hypotenuse is always the longest side, directly across from the right angle.

Solving right triangles

A right triangle has three sides and three angles. One angle is always 90 degrees. Since the angles must add up to 180 degrees, the other two angles must add up to 90 degrees. To solve a right triangle completely (find all six values) you need to know at least two pieces of information: either two sides, or one side and one angle (other than the right angle).

Example: A ladder 5 metres long leans against a wall at an angle of 65 degrees to the ground. How high up the wall does it reach? The ladder is the hypotenuse (5m), the angle at the ground is 65 degrees, and we want the opposite side (height). Using sin(65) = opposite/hypotenuse, height = 5 x sin(65) = 5 x 0.9063 = 4.53 metres.

The sine rule

For any triangle (not just right-angled ones), the sine rule states: a/sin(A) = b/sin(B) = c/sin(C). In plain English, each side divided by the sine of its opposite angle gives the same value. You use the sine rule when you know two angles and one side (AAS or ASA), or two sides and the angle opposite one of them (SSA, the ambiguous case).

Example: A surveyor needs to find the width of a river. From point A, the angle to a tree on the opposite bank is 58 degrees. From point B, 40 metres downstream, the angle to the same tree is 72 degrees. The angle at the tree (angle C) is 180 - 58 - 72 = 50 degrees. Using the sine rule: AB/sin(C) = AC/sin(B), so AC = 40 x sin(72) / sin(50) = 40 x 0.951 / 0.766 = 49.6 metres.

The cosine rule

The cosine rule is used when you know all three sides (SSS) or two sides and the angle between them (SAS). The formula is: c squared = a squared + b squared minus 2ab times cos(C). When C = 90 degrees, cos(C) = 0 and this simplifies to the Pythagorean theorem. You can also rearrange it to find an angle when you know all three sides: cos(C) = (a squared + b squared minus c squared) divided by 2ab.

Example: A ship travels 12km on a bearing of 050 degrees, then 9km on a bearing of 120 degrees. How far is it from its starting point? The bearing changes by 120 - 50 = 70 degrees, but the interior angle of the triangle at the turn vertex is the supplement of that turn: 180 - 70 = 110 degrees. Using the cosine rule: distance squared = 12 squared + 9 squared minus 2(12)(9)cos(110) = 144 + 81 plus 73.9 = 298.9. Distance = sqrt(298.9) = 17.3km.

Radians versus degrees

Degrees divide a full circle into 360 equal parts. Radians measure angles in terms of the radius of a circle. One full rotation is 2 pi radians (approximately 6.28). One radian is approximately 57.3 degrees. The conversion formulas are: radians = degrees x (pi/180) and degrees = radians x (180/pi). Radians become important in NCEA Level 3 calculus, where the derivatives of sin and cos are only clean (sin x gives cos x) when angles are in radians.