# Exact Value of tan 142½°

How to find the exact value of tan 142½° using the value of sin 15° and cos 15°?

Solution:

For all values of the angle A and B we know that, tan (A + B) = $$\frac{tan A + tan B}{1 - tan A tan B}$$,

sin A = 2 sin $$\frac{A}{2}$$ cos $$\frac{A}{2}$$

and

cos A = cos$$^{2}$$ $$\frac{A}{2}$$ – sin$$^{2}$$ $$\frac{A}{2}$$

Now tan 142½°

= tan (90 + 52½°)

= - cot 52½°

= $$\frac{-1}{tan 52½°}$$

= $$\frac{-1}{tan (45° + 7½°}$$

= - $$\frac{1 - tan 7½°}{1 + tan 7½°}$$

= - $$\frac{cos 7½° - sin 7½°}{cos 7½° + sin 7½°}$$

=  - $$\frac{(cos 7½° - sin 7½°)(cos 7½° - sin 7½°)}{(cos 7½° + sin 7½°)(cos 7½° - sin 7½°)}$$

= - $$\frac{(cos 7½° - sin 7½°)^{2}}{cos^{2} 7½° - sin^{2} 7½°}$$

= - $$\frac{1 - 2 sin 7½° cos 7½°}{cos^{2} 7½° - sin^{2} 7½°}$$

= - $$\frac{1 - sin 15°}{cos 15°}$$

= - $$\frac{1 - sin (45° - 30°)}{cos (45° - 30°)}$$

= - $$\frac{1 - \frac{\sqrt{3} - 1}{2\sqrt{2}}}{\frac{\sqrt{3} + 1}{2\sqrt{2}}}$$

= - $$\frac{2√2 - √3 + 1}{√3 + 1}$$

= - $$\frac{(2√2 - √3 + 1)}{(√3 + 1)}$$  $$\frac{(√3 - 1)}{(√3 - 1)}$$

= - $$\frac{(2√2 - √3 + 1)(√3 - 1)}{3 - 1}$$

= - $$\frac{(2√2(√3 - 1) - (√3 - 1)^{2}}{2}$$

= -[√2(√3 - 1) – (2 - √3)]

= -√6 + √2 + 2 - √3

= 2 + √2 - √3 - √6

Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.

## Recent Articles

1. ### Worksheet on Triangle | Homework on Triangle | Different types|Answers

Jun 21, 24 02:19 AM

In the worksheet on triangle we will solve 12 different types of questions. 1. Take three non - collinear points L, M, N. Join LM, MN and NL. What figure do you get? Name: (a)The side opposite to ∠L…

2. ### Worksheet on Circle |Homework on Circle |Questions on Circle |Problems

Jun 21, 24 01:59 AM

In worksheet on circle we will solve 10 different types of question in circle. 1. The following figure shows a circle with centre O and some line segments drawn in it. Classify the line segments as ra…

3. ### Circle Math | Parts of a Circle | Terms Related to the Circle | Symbol

Jun 21, 24 01:30 AM

In circle math the terms related to the circle are discussed here. A circle is such a closed curve whose every point is equidistant from a fixed point called its centre. The symbol of circle is O. We…

4. ### Circle | Interior and Exterior of a Circle | Radius|Problems on Circle

Jun 21, 24 01:00 AM

A circle is the set of all those point in a plane whose distance from a fixed point remains constant. The fixed point is called the centre of the circle and the constant distance is known