TRIGONOMETRIC FORMULAS
B
C
Right Triangle
A
Solution of Right Triangles
For Angle A. sin = \frac{a}{c}, cos = \frac{b}{c}, tan = \frac{a}{b}, cot = \frac{b}{a}, sec = \frac{c}{b}, cosec = \frac{c}{a}
Given Required
c
Oblique Triangles
a, b A, B, c tan A = \frac{a}{b} = cot B, c = \sqrt{a^2+b^2} = a\sqrt{1+\frac{b^2}{a^2}}
a, c A, B, b sin A = \frac{a}{c} = cos B, b = \sqrt{(c+a)(c-a)} = c\sqrt{1-\frac{a^2}{c^2}}
A, a B, b, c B=90^{\circ}-A, b=a cot A, c=\frac{a}{sin A}
A, b B, a, c B=90^{\circ}-A, a=b tan A, c=\frac{b}{cos A}
A, c B, a, b B=90^{\circ}-A, a=c sin A, b=c cos A
Solution of Oblique Triangles
Given Required
A, B, a b, c, C b = \frac{a sin B}{sin A}, C = 180^{\circ}-(A+B), c=\frac{a sin C}{sin A}
A, a, b B, c, C sin B = \frac{b sin A}{a}, C = 180^{\circ}-(A+B), c=\frac{a sin C}{sin A}
a, b, C A, B, c A+B=180^{\circ}-C, tan \frac{1}{2}(A-B)=\frac{(a-b)tan \frac{1}{2}(A+B)}{a+b},
c = \frac{a sin C}{sin A}
a, b, c A, B, C s=\frac{a+b+c}{2}, sin \frac{1}{2}A=\sqrt{\frac{(s-b)(s-c)}{bc}},
sin \frac{1}{2}B=\sqrt{\frac{(s-a)(s-c)}{ac}}, C=180^{\circ}-(A+B)
a, b, c Area s=\frac{a+b+c}{2}, area = \sqrt{s(s-a) (s-b) (s-c)}
A, b, c Area area = \frac{bc sin A}{2}
A, B, C, a Area area = \frac{a^2 sin B sin C}{2 sin A}
REDUCTION TO HORIZONTAL
Slope distance
Vert. Angle
Horizontal distance
Rise
Horizontal distance = Slope distance multiplied by the cosine of the vertical angle. Thus: slope distance=319.4 ft.
Vert. angle=5^{\circ} 10'. Since cos 5^{\circ} 10'=.9959, horizontal distance=319.4\times .9959=318.09 ft.
Horizontal distance also = Slope distance minus slope distance times (1-cosine of vertical angle). With the
same figures as in the preceding example, the following result is obtained. Cosine 5^{\circ} 10'=.9959.1-.9959=.0041.
319.4\times .0041=1.31. 319.4-1.31=318.09 ft.
When the rise is known, the horizontal distance is approximately the slope distance less the square of the rise divided by twice the slope distance. Thus: rise=14 ft.,
slope distance=302.6 ft. Horizontal distance=302.6-\frac{14 \times 14}{2 \times 302.6}=302.6-0.32=302.28 ft.