Find values of sin, cos, tan, sec, csc, and cot without a calculator

[Ryuk1990]

Homework Statement

I'm wondering how you would find values of sin, cos, tan, sec, csc, and cot without a calculator. I don't have a specific problem but how would you solve things like tan 45 degrees, sec 30 degrees, and cos -30 degrees?

2. Relevant info

I have the values of sin and cos memorized for 0, 30, 45, 60, 90, 180, and 270 degrees. There is a technique to solving the problems above knowing whether the trig functions are positive or negative in specific quadrants. How is that relevant to solving for the values?

The Attempt at a Solution

I know how to solve some of the above using identities. For example, tan 45 is just sin 45/cos 45 which is 1. For sec 30, I believe it's just 1/cos 45 so it'd be 1/(\sqrt{2}/2).

I don't know how to solve cos -30. How would you solve the problems without identities? There is a way knowing when the functions are positive/negative in the quadrants. It also has something to do with adding and subtracting the angle measurement. Can someone explain the technique please?

 

[tiny-tim]

Hi Ryuk1990!

(have a square-root: √ and a degree: º and a theta: θ )

Use sin = opp/hyp, cos = adj/hyp, tan = opp/adj, plus the fact that a 45º triangle is half a square, and a 30º or 60º triangle is half an equilateral triangle.

(but sec30º = 1/cos30º, of course)

I don't know how to solve cos -30. How would you solve the problems without identities? There is a way knowing when the functions are positive/negative in the quadrants. It also has something to do with adding and subtracting the angle measurement. Can someone explain the technique please?

Personally, I always use the formula for cos(180º ± θ), also cos(-θ) = cosθ, sin(-θ) = -sinθ.

But you can also do it by drawing the angle on a graph, and using x = rcosθ, y = rsinθ (so eg in the second quadrant, x is negative but y is positive, so cos is negative but sin is positive).