Question

IF a+b=90 degree AND b+c=a THEN WHAT IS THE VALUE OF tan a. DO TRY!!!

Answer 1

IF a+b=90 degree AND b+c=a THEN WHAT
IS THE VALUE OF tan a.
IF a+b=90 degree AND b+c=a THEN WHAT
IS THE VALUE OF tan a. DO TRY!!!

Before I start, I must say I have found no
mathematical way to settle on just one set of values.

a + b = 90        b + c = a
b = 90 - a         (90 - a) + c = a
                       90 + c = 2a
(b + c) + b = 90
2b + c = 90

We can keep manipulating these, but there
is no way to eliminate one of the variables so
we can zero in on another. So, let's make a table.

a	b	c	2b	a+b	90+c	2a
89	1	88	2	90	178	178
88	2	86	4	90	176	176
87	3	84	6	90	174	174
86	4	82	8	90	172	172
85	5	80	10	90	170	170
84	6	78	12	90	168	168
83	7	76	14	90	166	166
82	8	74	16	90	164	164
81	9	72	18	90	162	162

We can extend that up to the point where b = 45,

49	41	8	82	90	98	98
48	42	6	84	90	96	96
47	43	4	86	90	94	94
46	44	2	88	90	92	92
45	45	0	90	90	90	90

It won't do any good to go beyond that because c becomes negative.

Formual 2b + c = 90. Add the entries in cols 3 and 4. Always works.
Formula 90 + c = 2a. Compare entries in cols 6 and 7. Always works.
Formula b + c = a. Add entries in cols 2 and 3, compare to col 1. Always works.
Formula a + b = 90. Add entries in cols 1 and 2. Always works.

The best we can do is decide "I want symmetry." So we choose 60 degrees for
a and 30 degrees for both b and c. Try them in every formula above; they work.

Using those choices, the tangent of a is 1.732.

Answer 2

If a+b=90 and b+c=a, then 2b+c=90. tan(2b)=tan(90-c)=cot(c)=2tan(b)/(1-tan²(b)). So tan(c)=(1-tan²(b))/(2tan(b)). If t=tan(b), tan(c)=(1-t²)/(2t), t²+2ttan(c)=1, (t+tan(c))²=1+tan²(c)=sec²(c), t=tan(b)=sec(c)-tan(c)=(1-sin(c))/cos(c).

tan(a)=tan(90-b)=cot(b)=1/tan(b); tan(a)=cos(c)/(1-sin(c)).

With only two equations and three unknowns, no actual value for tan(a) can be calculated, so it's assumed that tan(a) must be expressed in terms of either b or c.