Question

Is there a way to calculate sin, cos, and tan without a calculator?

If so, could you please do it with this problem...

Tan(90°) = DE/4

Also, could you please explain it step by step.

Thanks!

Answer 1

See below.

Explanation:

You can find them by constructing a right angled triangle with the given angle and measuring the sides giving sin = opposite / hypotenuse , cos = adjacent / hypotenuse and tan = opposite / adjacent ( SOH-CAH-TOA).

For tan 90 degrees you think of an 'imaginary' triangle where opposite side = hypotenuse and the adjacent side = 0 ( in fact a vertical line!) .Now tan x = opposite / adjacent = opposite / 0 which is indeterminate. There is no value for the tan of 90 degrees and so there is no solution to your problem.

Answer 2

yes, and no

Explanation:

There are a few trig functions that are "simple" and can be memorized. Usually, these are the ones for angles of 0°, 30°, 45°, 60°, 90°. See below for a table.

To solve an equation like this for DE, you multiply both sides of the equation by 4 (cancelling the denominator).

4·tan(90°) = DE

The tangent of 90° is "infinity." So, the length of DE will be infinite for an angle of 90°.

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Trig functions of multiple angles and half angles can be derived from those in the table using the appropriate formulas. The math can get messy fast.

Of course, SOH CAH TOA reminds you of the relationships between trig function values and right triangle side measures. You may not know the angle measure (and it may be irrelevant), but you can find the trig function value.