Question

How do you evaluate sine, cosine, tangent of \[\dfrac{\pi }{4}\] without using a calculator?

Answer 1

Hint:In the above question, is based on the concept of trigonometry. The sine, cosine, tangent functions can be solved by solving the relationship between angles and sides of the triangle.

Using \[\dfrac{\pi }{4}\] as the angle we can find the length of sides to get the value of each function.

Complete step by step solution:
Given is the angle which is \[\dfrac{\pi }{4}\] or \[{45^ \circ }\].So to solve the value of functions, we
need to know that one angle in the triangle is always \[{90^ \circ }\].Since we want to find the function for the angle for \[{45^ \circ }\],so the other angle in the triangle is \[{45^ \circ }\].

The total angle of triangle is \[{180^ \circ }\].Therefore by adding two angles and subtracting it with
\[{180^ \circ }\] we get a third angle.
\[{180^ \circ } - ({90^ \circ } + {45^ \circ }) = {45^ \circ }\].

So, we consider the unit triangle \[\vartriangle ABC\]where two sides are equal. Since the two sides of triangles are equal, the triangle is an isosceles triangle. Therefore, by applying Pythagoras theorem, we
get \[AC = \sqrt {A{B^2} + B{C^2}} = \sqrt {{1^2} + {1^2}} = \sqrt 2 \]

Now, we have to apply the trigonometric properties for different functions. For sine function the formula is the opposite side divided by hypotenuse.
\[\sin {45^ \circ } = \dfrac{1}{{\sqrt 2 }}\]

For cosine function, the formula is adjacent side divided by hypotenuse. Therefore, we get
\[\cos {45^ \circ } = \dfrac{1}{{\sqrt 2 }}\]

For tangent function, the formula is sine function divided by cosine function. \[\]
\[\tan {45^ \circ } = \dfrac{{\dfrac{1}{{\sqrt 2 }}}}{{\dfrac{1}{{\sqrt 2 }}}} = 1\]

Therefore, we get the above values.

Note: An important thing to note is that \[\dfrac{1}{{\sqrt 2 }}\] can also be further solved to get a decimal numbers.So, rationalizing it with \[\sqrt 2 \],we get \[\dfrac{1}{{\sqrt 2 }} \times \dfrac{{\sqrt 2 }}{{\sqrt 2 }} = \dfrac{{\sqrt 2 }}{2}\].The value of \[\sqrt 2 \]is 1.414,therefore dividing it by 2 we get
0.707.