Finding the exact value of Inverse Trig Functions without a calculator

Hey I'm in 11th grade in Advanced Trigonometry and need help with Inverse Trig Functions,

**Find the exact value of each expression without calculator.**

sin [arctan (-5/12)]

cot [arccsc(-9/4)]

-Those are examples of the problems I need to solve any help would be greatly appreciated

Thanks,

Sean

Re: Finding the exact value of Inverse Trig Functions without a calculator

If $\displaystyle \theta= arctan(-5/12)$ then $\displaystyle tan(\theta)= -5/12$= "opposite side"/"near side'. Draw a right triangle with one vertex at (0, 0), another at (-12, 0), and the third at (-12, 5). Label the angle at the origin "$\displaystyle \theta$". On that triangle $\displaystyle tan(\theta)= -5/12$. What is $\displaystyle sin(\theta)$?

If $\displaystyle \theta =arccsc(-9/4)$, then [tex]csc(\theta)= -9/4[//tex]= "hypotenuse"/"opposite side". In order to draw such a triangle, we need to know the length of the "near side". Use the Pythagorean theorem.

Re: Finding the exact value of Inverse Trig Functions without a calculator

Quote:

Originally Posted by

**srob22** Hey I'm in 11th grade in Advanced Trigonometry and need help with Inverse Trig Functions,

**Find the exact value of each expression without calculator.**

sin [arctan (-5/12)]

cot [arccsc(-9/4)]

-Those are examples of the problems I need to solve any help would be greatly appreciated

Thanks,

Sean

What is the 3rd side in the right triangle with sides -5 and 12? -9 and 4? Identify the angle arctan and arccsc refer to and calculate the ratios for sin and cot in each respective triangle.

Re: Finding the exact value of Inverse Trig Functions without a calculator

I missed the previous replies, but here goes.

Quote:

Originally Posted by

**srob22** sin [arctan (-5/12)]

Draw the unit circle and an angle whose tangent is -5/12. Find the right triangle with hypotenuse 1 that is similar to the right triangle with legs 5 and 12. The smaller of those legs is the absolute value of sin [arctan (-5/12)]. For the sign, consider which of the two angles with tangent -5/12 is in the domain of arctangent.

Alternatively, express sine through tangent using these formulas.

Quote:

Originally Posted by

**srob22** cot [arccsc(-9/4)]

First, arccsc(x) = arcsin(1/x) and arcsin(-x) = -arcsin(x). Then again, either find the right triangle with hypotenuse 1 and one leg 4/9 and find the ratio of the legs, or express cotangent through sine.

Re: Finding the exact value of Inverse Trig Functions without a calculator

Quote:

Originally Posted by

**HallsofIvy** If $\displaystyle \theta= arctan(-5/12)$ then $\displaystyle tan(\theta)= -5/12$= "opposite side"/"near side'. Draw a right triangle with one vertex at (0, 0), another at (-12, 0), and the third at (-12, 5). Label the angle at the origin "$\displaystyle \theta$". On that triangle $\displaystyle tan(\theta)= -5/12$. What is $\displaystyle sin(\theta)$?

Yes, though the final result should be negative.