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Strictly speaking, we talk about the sine, cosine and tangent of angles not triangles. A typical problem setting is that you are given one of the angles and one of the sides in a right-angled triangle.
First, in order to understand trig, you should know its basics! Trig consists of three angles: \(Sin\), \(Cos\), and \(Tan\). What do these mean? Sin is the angle measured to the opposite side of the hypotenuse where \(sinx=opp/hyp\). Cos is the angle measured to the adjacent side of the hypotenuse where \(cosx=adj/hyp\).
- Motivation: Trig Is Anatomy
- Sine/Cosine: The Dome
- Tip: Trig Values Are Percentages
- Tangent/Secant: The Wall
- Cotangent/Cosecant: The Ceiling
- Visualize The Connections
- Gotcha: Remember Other Angles
- Summary: What Should We Remember?
- Appendix: The Original Definition of Tangent
- Appendix: Inverse Functions
Imagine Bob The Alien visits Earth to study our species. Without new words, humans are hard to describe: “There’s a sphere at the top, which gets scratched occasionally” or “Two elongated cylinders appear to provide locomotion”. After creating specific terms for anatomy, Bob might jot down typical body proportions: 1. The armspan (fingertip to fing...
Instead of staring at triangles by themselves, like a caveman frozen in ice, imagine them in a scenario, hunting that mammoth. Pretend you’re in the middle of your dome, about to hang up a movie screen. You point to some angle “x”, and that’s where the screen will hang. The angle you point at determines: 1. sine(x) = sin(x) = height of the screen, ...
Nobody ever told me in my years of schooling: sine and cosine are percentages. They vary from +100% to 0 to -100%, or max positive to nothing to max negative. Let’s say I paid $14 in tax. You have no idea if that’s expensive. But if I say I paid 95% in tax, you know I’m getting ripped off. An absolute height isn’t helpful, but if your sine value is...
One day your neighbor puts up a wall right nextto your dome. Ack, your view! Your resale value! But can we make the best of a bad situation? Sure. What if we hang our movie screen on the wall? You point at an angle (x) and figure out: 1. tangent(x) = tan(x) = height of screen on the wall 2. distance to screen: 1 (the screen is always the same dista...
Amazingly enough, your neighbor now decides to build a ceiling on top of your dome, far into the horizon. (What’s with this guy? Oh, the naked-man-on-my-wall incident…) Well, time to build a ramp to the ceiling, and have a little chit chat. You pick an angle to build and work out: 1. cotangent(x) = cot(x) = how far the ceiling extends before we con...
A short time ago I had zero“intuitive conclusions” about the cosecant. But with the dome/wall/ceiling metaphor, here’s what we see: Whoa, it’s the same triangle, just scaled to reach the wall and ceiling. We have vertical parts (sine, tangent), horizontal parts (cosine, cotangent), and “hypotenuses” (secant, cosecant). (Note: the labels show where ...
Psst… don’t over-focus on a single diagram, thinking tangent is always smaller than 1. If we increase the angle, we reach the ceiling before the wall: The Pythagorean/similarity connections are always true, but the relative sizes can vary. (But, you might notice that sine and cosine are always smallest, or tied, since they’re trapped inside the dom...
For most of us, I’d say this is enough: 1. Trig explains the anatomy of “math-made” objects, such as circles and repeating cycles 2. The dome/wall/ceiling analogy shows the connections between the trig functions 3. Trig functions return percentages, that we apply to our specific scenario You don’t need to memorize 12+cot2=csc2, except for silly tes...
You may see tangent defined as the length of the tangent line from the circle to the x-axis (geometry buffs can work this out). As expected, at the top of the circle (x=90) the tangent line can never reach the x-axis and is infinitely long. I like this intuition because it helps us remember the name “tangent”, and here’s a nice interactive trig gui...
Trig functions take an angle and return a percentage. sin(30)=.5means a 30-degree angle is 50% of the max height. The inverse trig functions let us work backwards, and are written sin−1 or arcsin (“arcsine”), and often written asinin various programming languages. If our height is 25% of the dome, what’s our angle? Plugging asin(.25) into a calcul...
Learn and revise trigonometric ratios of sine, cosine and tangent and calculate angles and lengths in right-angled triangles with GCSE Bitesize AQA Maths.
Try Sin Cos and Tan. Play with this for a while (move the mouse around) and get familiar with values of sine, cosine and tangent for different angles, such as 0°, 30°, 45°, 60° and 90°.
Edexcel. Trigonometry - Edexcel Trigonometric ratios. The three trigonometric ratios; sine, cosine and tangent are used to calculate angles and lengths in right-angled triangles. The sine and...
People also ask
How do the sine and cosine rules work?
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Use cosine, sine and tan to calculate angles and sides of right-angled triangles in a range of contexts.
