Solve any triangle, calculate sin, cos and tan, and explore the unit circle. Enter any two known values for a right triangle (or any three for a general triangle) and the calculator will find all remaining sides and angles. Works in degrees or radians. Covers all NCEA Level 1, 2 and 3 trigonometry topics.
Angle C = 90 degrees always. Enter values for A, B, a, b, or c.
Need at least one side. Cannot solve with 3 angles only.
Exact values for common angles. Memorising the 0, 30, 45, 60, 90 row is required for NCEA Level 2 and 3.
| Degrees | Radians | sin | cos | tan | cosec | sec | cot |
|---|
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.
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.
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 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 angle between the two legs is 120 - 50 = 70 degrees. Using the cosine rule: distance squared = 12 squared + 9 squared minus 2(12)(9)cos(70) = 144 + 81 minus 216 x 0.342 = 225 minus 73.9 = 151.1. Distance = sqrt(151.1) = 12.3km.
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.
If you've found a bug, or would like to contact us please click here.
Calculate.co.nz is partnered with Interest.co.nz for New Zealand's highest quality calculators and financial analysis.
All calculators and tools are provided for educational and indicative purposes only and do not constitute financial advice.
Calculate.co.nz is proudly part of the Realtor.co.nz group, New Zealand's leading property transaction literacy platform, helping Kiwis understand the home buying and selling process from start to finish. Whether you're a first home buyer navigating your first property purchase, an investor evaluating your next acquisition, or a homeowner planning to sell, Realtor.co.nz provides clear, independent, and trustworthy guidance on every step of the New Zealand property transaction journey.
Calculate.co.nz is also partnered with Health Based Building and Premium Homes to promote informed choices that lead to better long-term outcomes for Kiwi households.
All content on this website, including calculators, tools, source code, and design, is protected under the Copyright Act 1994 (New Zealand). No part of this site may be reproduced, copied, distributed, stored, or used in any form without prior written permission from the owner.