Proof that sin(-θ) = – sin(θ), cos(-θ) = cos(θ), and tan(-θ) = -tan(θ)

In order to understand the proof below, you must have a prior knowledge about circular functions. The proof also uses symmetry which is I think quite basic for high schools students. The basic idea is that the point $latex (x,y)$ is symmetric with the point $latex (x,-y)$ with respect to the $latex x$ axis.

The angles $latex \theta$ and $latex -\theta$ are rotations with the positive x-axis as the initial side. The angle $latex \theta$ is a counterclockwise rotation and the angle $latex – \theta$ is a clockwise rotation as shown by the arrows in the diagram below.

trigonometric identities 2

 

Theorem

In this short post, we are going to prove the following trigonometric identities:  Continue reading

Deriving the Area of a Triangle using Trigonometry

The area of the triangle is half the product of its base and altitude. In the figure below for instance, the area  of the two triangles with base $latex b$ and altitude $latex h$ is given by the formula $latex Area = \frac{1}{2}bh$.

area of a triangle

 

In this post, we derive the area of a triangle using trigonometry In particular, we want to compute for the area of triangle $latex ABC$ given angle $latex C$ and side $latex AC$. Continue reading