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Trigonometry

Preface

Trigonometry is not about triangles, it’s all about circles.

Slope

Represents how a line rises or falls.

Radian

A radian starts from 180 degree (far left x, aka -x) in the second quadrant and goes clockwise.

π\pi is the ratio of a circle’s circumference to its diameter, approximately 3.14159.

There are 2π2\pi radians in a full circle, where

π3 radians= 60 degrees \dfrac{\pi}{3} \text{ radians} = \text{ 60 degrees } π radians= 180 degrees \pi \text{ radians} = \text{ 180 degrees } 2π radians= 360 degrees 2\pi \text{ radians} = \text{ 360 degrees }

It represents the ratio of the circumference of a circle to its radius.

Since we have the ratio we can use the radius to find the actual value of the circumference w/

C=2πrC = 2\pi r

And we can find the area of it by

Area =length  breadth=πrr=πr2 \begin{split} \text{Area } &= \text{length } \cdot \text{ breadth} \\ &= \pi r \cdot r \\ &= \pi r^2 \end{split}

Sin & Cos

sin(0) is measuring the distance of y and cos(0) is measuring the distance of the x coordinate as you walk around the circle in an anti-clockwise fashion.

both will inevitably have the exact same graph because they cycle through the quadrants in the same manner, just delayed. However, since they start at different points, y and x, it means the graphs are translated. So if we started from the top of the circle at our y then x = 0 whereas if we started at far right for x then y = 0. So there is a translation of 1.

When they go from 1 to -1 what is really happening is the trough of the graph is peaking but inverted since turning it into a negative flips the graph.

If we think of the question: what would the sin and cos be if our radian was 3 in a circle?

Well we know x = cos and y = sin so as we start from the far

cos(3)0.99sin(3)0.14\begin{split} \cos(3) &\approx -0.99 \\ \sin(3) &\approx 0.14 \end{split}

Tan

tan is a shorterned name from tangent because the point of it is where the slope is tangent to the point on the circumference point of the circle.

tan(0) is measuring the distance of the radius, that is made up from sin(0) and cos(0). And since the circle’s points are x = 1 and -x = -1 the set of all solution points is the graph of the equation.