Practice for finding the other trig functions and the angle from the value of sine, cosine or tangent, assuming all are positive

Example #1: cosalpha = 2/5

(2/5)² + sin²alpha = 1
4/25 + sin²alpha = 1
sin²alpha = 21/25
sinalpha = sqrt(21)/5

since we have sine and cosine, tan = sin/cos.

tanalpha = [sqrt(21)/5]/[2/5] = sqrt(21)/2

With this editor, I can't write -1 as a superscript, so instead I will use the words arcsine, arccosine and arc tantangent. On your calculator, arccos(2/5) = 66.42182152...°, which we round to 66.4218°.

If you take the unrounded value and change it to degrees-minutes-seconds, rounding the seconds to the nearest whole number, we get 66° 25' 19".

Example #1.1: If we had sinalpha = 2/5, the work would look nearly identical, except cosalpha would equal sqrt(21)/5.  Tangent of this angle is the reciprocal of sqrt(21)/2, which is 2sqrt(21)/21 when written in rational denominator form.  The angle is the complement of our original angle, which means they add up to 90°. arcsin(2/5) = 23.57817848...°, which rounds to 23.5782°.

The DMS version of the unrounded value, rounded to the nearest whole second is 23° 34' 41".

Example #2: tanbeta = 2/5

Since tan = sin/cos,
tan × cos = sin

Using this, cos²beta + (2/5)²cos²beta = 1

cos²beta + 4/25cos²beta = 1
29/25cos²beta = 1
cos²beta =25/29
cosbeta = sqrt(25/29) = 5/sqrt(29) = 5sqrt(29)/29

arctan(2/5) = 21.80140949...°, which rounds to 21.8014°, and in DMS is 21° 48' 5".

Here are practice problems. Get the other two trig values, the angle rounded to the nearest ten thousandth of a degree and rounded to the nearest second in DMS mode.

1) cosalpha = 1/10

2) tanbeta = 1/10

3) singamma = 1/9

4) tandelta = 1/9

Answers in the comments.