Machinery's Handbook, 31st Edition
96 FUNCTIONS OF ANGLES Referring to the figure at the bottom of page 95, the ratios of the sides of a right trian- gle with respect to angle A are named sine A , cosine A, and tangent A (abbreviated sin A , sin A , and tan A ). Ratios and reciprocal ratios, named cosecant A , secant A, and cotangent A (abbreviated csc A , sec A , and cot A ) are defined as:
opposite hypotenuse -------------- adjacent hypotenuse --------------
hypotenuse opposite -------------- hypotenuse adjacent --------------
1 A sin ------ 1 A cos ------
a c --
c a --
A sin
=
=
A csc
=
= =
c b --
b c --
A sec
A cos
=
=
=
= =
adjacent opposite ----------
opposite adjacent ----------
1 A tan ------
b a --
a b --
A cot
A tan
=
=
=
= =
Similar ratios are defined for angle B :
opposite hypotenuse -------------- adjacent hypotenuse --------------
hypotenuse opposite -------------- hypotenuse adjacent --------------
1 B sin ------ 1 B cos ------
b c --
c b --
B sin
=
=
B csc
=
= =
c a --
a c --
B sec
B cos
=
=
=
= =
adjacent opposite ----------
opposite adjacent ----------
1 B tan ------
a b --
b a --
B cot
B tan
=
=
=
= =
Thus, in a given right triangle, sin A = cos B , cos A = sin B , tan A = cot B . Similarly, csc A = sec B , sec A = csc B , cot a = tan B . Law of Sines.— In any triangle, if a , b , and c are the sides, and A , B , and C their opposite angles, respectively, then: a
c C sin ------
b B sin ------
A sin ------
=
=
so that:
a c A sin C sin = ------- b c B sin C sin = -------
a b A sin B sin = -------- b a B sin A sin = -------- c a C sin A sin = --------
or
or
or c b C sin B sin = -------- Law of Cosines.— In any triangle, if a , b and c are the sides and A , B , and C are the opposite angles, respectively, then: a 2 b 2 c 2 2 bc A cos = + – b 2 a 2 c 2 2 ac B cos = + – c 2 a 2 b 2 2 ab C cos = + – The sine and cosine laws together with the proposition that the sum of the measures of the three angles is 180 degrees are the basis of all formulas relating to the solution of triangles. Formulas and examples for the solution of right-angle and oblique-angle triangles, ar- ranged in tabular form, are given on the following pages. Trigonometric Identities.— It is possible to express trigonometric ratios in terms of other ratios by way of trigonometric identities. For example, sin( A + B ) = sin A cos B + cos A sin B . It may be helpful to use an identity to quickly evaluate a trigonometric function, as shown by the examples given below the trigonometric identities and formulas derived from them.
Copyright 2020, Industrial Press, Inc.
ebooks.industrialpress.com
Made with FlippingBook - Share PDF online