Solution of Right Triangles Solution of Right Triangles Machinery's Handbook, 31st Edition
98
Right triangles (one angle measures 90°) are the easiest to “solve” (that is, to find the missing sides or angles), thanks to the Pythagorean theorem and right triangle ratios.
The figure to the left shows right triangle ABC . Sides opposite corresponding angles are labeled a , b , c . The formulas in the table are for finding an unknown side or angle from given information as shown. There are several ways to solve for the missing dimension through one of the three basic trigonometric functions. Right angle C is always known (90°), thus, A = 90 – B and B = 90 – A .
B
c
a
C
90°
A
b
Sides and Angle Known
Formulas for Sides, Angles to be Found
Sides a and b
B = 90 ° - A
A = tan –1 ( a / b )
a 2 b 2 = +
c
A = sin –1 ( a / c ) or A = cos –1 ( b / c ) B = sin –1 ( b / c ) or B = cos –1 ( a / c )
Side a , hypotenuse c
B = 90 ° - A
b c 2 a 2 = –
Side b , hypotenuse c
A = 90 ° - B
a c 2 b 2 = –
b = c sin B or a = c cos B b = c cos A or a = c sin A
a = c cos B or b = c sin B a = c sin A or b = c cos A
Hypotenuse c , angle B
A = 90 ° - B
Hypotenuse c , angle A
B = 90 ° - A
b B sin = ------ a B cos = ------ or b A cos = ------ A sin = ------ a or a B cos = ------ a A sin = ------
c
Side b , angle B
A = 90 ° - B
B tan = ------
a b
c
c
Side b , angle A
a = b tan A
B = 90 ° - A
c
Side a , angle B
b = a tan B
A = 90 ° - B
c
Side a , angle A B = 90 ° - A The solutions of triangles—both right and oblique—rely not only on the Pythagorean theorem and the laws of cosines and of sines, but also on correct use of the trigonometry c b a A tan = ------
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