Example 12 Prove that
Solution
Example 13
Verify that
Solution
Example 14 If x = r cos A cos
B, y= r cos A sin B and z = r sin A prove that X^{2} + Y^{2}
+ z^{2} = r^{2}
Solution
x^{2} + y^{2}+ z^{2}=
r^{2} ......(given to be proved.)
x^{2} + y^{2}+ z^{2}= r^{2}
\ (r cos A cos B)^{2}
+ (r cos A sin B)^{2}+
(r sinA) ^{ 2} = r ^{ 2} ....substituting the given
relations.
\ r^{2}cos^{2}A
cos^{2 }B + r^{2}cos^{2} A sin^{2}
B + r^{2}sin^{2}A = r^{2} ....(algebraic
manipulation)
\ r^{2}cos ^{2}
A (cos^{2}B+ sin^{2}B) + r^{2} sin^{2}A
= r^{2} .....(combining terms)
\ r^{2}cos^{2}A
(1) + r^{2}sin^{2} A = r^{ 2 } ....Pythagorean
identity
\ r^{2}(cos^{2}
A + sin^{2} A) = r^{2}
\ r^{2}(1) .....Pythagorean
identity.
Example 15
Solution
Example 16 Show that
Solution
We know that
since (xy)^{2}is a complete square, it
can't be negative.
\ (x y)^{2}=
0 is the only possibility.
\ (x y) = 0 i.e. x
= y.
Example 17 If a and b are two
positive real numbers. Can
ii) tan x = b/a? Give reasons.
Solution
Example 18 Express each of
other trigonometric ratios of
in terms of sin .
Solution
Example 19 If
is in Quadrant IV, express the other trigonometric functions of
in terms of sec .
Solution
[next page]
