Question

How do you find the exact value of the six trigonometric functions of ${{45}^{\circ }}$?

Answer 1

We first express all the six trigonometric functions. We divide them in primary ratios and their inverse ratios. We also find all possible relations between those ratios. Then we take the angle values of ${{0}^{\circ }}$ for all the six trigonometric functions.

Complete step by step solution:
We first complete the list of all the six trigonometric functions.
The main three trigonometric ratio functions are $\sin \theta ,\cos \theta ,\tan \theta$. The inverse of these three functions is $\csc \theta ,\sec \theta ,\cot \theta$. Also, we can express $\tan \theta =\dfrac{\sin \theta }{\cos \theta }$.
Therefore, the relations are $\csc \theta =\dfrac{1}{\sin \theta },\sec \theta =\dfrac{1}{\cos \theta },\cot \theta =\dfrac{1}{\tan \theta }$.
We can also express these ratios with respect to a specific angle $\theta$ of a right-angle triangle and use the sides of that triangle to find the value of the ratio.
A right-angle triangle has three sides and they are base, height, hypotenuse. We express the ratios in $\sin \theta =\dfrac{\text{height}}{\text{hypotenuse}},\cos \theta =\dfrac{\text{base}}{\text{hypotenuse}},\tan \theta =\dfrac{\text{height}}{\text{base}}$.
Similarly, $\csc \theta =\dfrac{\text{hypotenuse}}{\text{height}},\sec \theta =\dfrac{\text{hypotenuse}}{\text{base}},\cot \theta =\dfrac{\text{base}}{\text{height}}$.
Now we express the values of these ratios for the conventional angles of ${{45}^{\circ }}$.

Ratios	angles (in degree)	values
$\sin \theta$	${{45}^{\circ }}$	$\dfrac{1}{\sqrt{2}}$
$\cos \theta$	${{45}^{\circ }}$	$\dfrac{1}{\sqrt{2}}$
$\tan \theta$	${{45}^{\circ }}$	1
$\csc \theta$	${{45}^{\circ }}$	$\sqrt{2}$
$\sec \theta$	${{45}^{\circ }}$	$\sqrt{2}$
$\cot \theta$	${{45}^{\circ }}$	1

Note: We need to remember that in mathematics, the trigonometric functions are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others.