Question

answer the following questions using what youve learned from this unit. write your responses in the space provided.

1. complete the empty cells in the table below with the correct values. (2 points)

$\theta$ in degrees	$\theta$ in radians	$\sin\theta$	$\cos\theta$	$\tan\theta$	$\csc\theta$	$\sec\theta$	$\cot\theta$

Explanation:

Step1: Recall cosine value

We know that \(\cos\theta = 1\). From the unit circle, we remember that \(\cos\theta=1\) when \(\theta = 0^{\circ}\) (or \(\theta = 0\) radians, since \(0^{\circ}\) in radians is \(0\times\frac{\pi}{180}=0\) radians).

Step2: Calculate other trigonometric values

• For \(\sin\theta\): Using the identity \(\sin^{2}\theta+\cos^{2}\theta = 1\), substitute \(\cos\theta = 1\). Then \(\sin^{2}\theta+1^{2}=1\), so \(\sin^{2}\theta=0\), which means \(\sin\theta = 0\).
• For \(\tan\theta\): The formula for tangent is \(\tan\theta=\frac{\sin\theta}{\cos\theta}\). Substituting \(\sin\theta = 0\) and \(\cos\theta = 1\), we get \(\tan\theta=\frac{0}{1}=0\).
• For \(\csc\theta\): The cosecant function is the reciprocal of sine, i.e., \(\csc\theta=\frac{1}{\sin\theta}\). But \(\sin\theta = 0\), and division by zero is undefined, so \(\csc\theta\) is undefined.
• For \(\sec\theta\): The secant function is the reciprocal of cosine, i.e., \(\sec\theta=\frac{1}{\cos\theta}\). Substituting \(\cos\theta = 1\), we get \(\sec\theta=\frac{1}{1}=1\).
• For \(\cot\theta\): The cotangent function is the reciprocal of tangent, i.e., \(\cot\theta=\frac{\cos\theta}{\sin\theta}\). But \(\sin\theta = 0\), and division by zero is undefined, so \(\cot\theta\) is undefined.

Answer:

\(\boldsymbol{\theta}\) in degrees	\(\boldsymbol{\theta}\) in radians	\(\boldsymbol{\sin\theta}\)	\(\boldsymbol{\cos\theta}\)	\(\boldsymbol{\tan\theta}\)	\(\boldsymbol{\csc\theta}\)	\(\boldsymbol{\sec\theta}\)	\(\boldsymbol{\cot\theta}\)