In this post, I find an expression for the radial unit vector, $\vec{e}_r$. The three unit vectors in the following digram form a right-handed spherical coordinate system. This unit vector is easier to find than the other two unit vectors because all that is needed is vector addition. The Radial Unit Vector in Terms of…

# Category: Trigonometry

## Proof of the Triangle Inequality for Real Numbers

The triangle inequality for real numbers is $ |a+b| \le |a| + |b|$ in which $a$ is a variable for a real number, and $b$ is a variable for a real number. Proof: I use four cases. Case 1: If $a=0$ and $b$ is any real number, then the left side of the triangle inequality…

## Do the angle addition identities only work for positive angles?

The Angle Addition Identities were derived using $\sin\theta \equiv \frac{opp}{hyp}$ and $\cos\theta \equiv \frac{adj}{hyp}$. In addition, $\theta$ was constrained to the interval of $[0, \frac{\pi}{2})$. The question addressed in this post is, do the Angle Addition Identities work if the arguments of the sine function and cosine functions are generalized to include negative numbers? In the following,…

## The Pythagorean Theorem and the Unit Circle

This following diagram shows that $\displaystyle \sin^2 \theta + \cos^2 \theta = 1$, given $ \displaystyle \cos \theta \equiv \frac{adj}{hyp}$ and $ \displaystyle \sin \theta \equiv \frac{opp}{hyp}$ and the Pythagorean theorem. Here, $opp$, $hyp$, and $adj$ are variables for lengths. The values corresponding to these lengths are positive numbers or zero.