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Study Notes 039:
Trig. Summary 13:
By William E. Steinman:
The main thing I want to do here is to bring all of our trig functions and identities onto a single page. That makes reference much easier.
We can begin with our six trigonometric functions, which are not derived, but given, based on the properties of a right triangle.
The sin of an angle is = to the side opposite/the hypotenuse.
The cosine of an angle is = to the side adjacent/the hypotenuse.
The tangent of an angle is = to the side opposite/the side adjacent.
The cosecant of an angle is = to the hypotenuse/the side opposite.
The secant of an angle is = to the hypotenuse/the side adjacent.
The cotangent of an angle is = to the side adjacent/the side opposite.
Notice that the cosecant, the secant, and the cotangent are the reciprocals of the sin, cosine, and tangent, respectively.
That is, given the angle A:
csc A = 1/sin A
sec A = 1/cos A
cot A = 1/tan A
We can also see that:
tan A = sin A /cos A
cot A = cos A/sin A
From these basic relationships, the trigonometric tables can be computed to any degree of accuracy required.
Also from these basic relationships we were able to derive the fundamental identities.
sin2 A + cos2 A = 1
1 + tan2 A = sec2 A
1 + cot2 A = csc2 A
tan A = sin A/cos A
You proved the formulas for addition and subtraction of two angles in SN032.
sin (a + b) = sin a cos b + cos a sin b
cos (a + b) = cos a cos b – sin a sin b
tan (a + b) = (tan a + tan b)/(1 – tan a tan b)
Sin (a - b) = sin a cos b – cos a sin b
cos (a – b) = cos a cos b + sin a sin b
tan (a – b) = (tan a – tan b)/1 + tan a tan b
We also came upon the double and half angle formulas.
sin 2a = 2 sin a cos b
cos 2a = cos2 a – sin2 b
tan 2a = (2tan a)/ (1- tan2 a)
We also have products, sums and differences of two angles.
sin a cos b = ½(sin (a + b) + sin(a - b))
cos a sin b = ½(sin (a + b) – sin(a – b))
cos a cos b = ½(cos (a + b) + cos(a – b))
sin a sin b = -½(cos (a + b) – cos(a – b))
sin a + sin b = 2 sin ½(a + b) cos ½(a - b)
sin a – sin b = 2 cos ½(a + b) sin ½(a – b)
cos a + cos b = 2cos ½(a + b) cos ½(a – b)
cos a – cos b = -2sin ½(a + b) sin ½(a – b)
Now, we have the laws of sins.
This is true for any triangle.
Given a triangle with sides A, B, and C with opposite angles of a, b, and c.
a/sin A = b/sin B = c/sin C
That is equivalent to sin A/a = sin B/b = sin C/c.
We also have the laws of cosines for the same triangle.
a2 = b2 + c2 – 2bc cos A
b2 = a2 + c2 – 2ac cos B
c2 = a2 + b2 – 2ab cos C
This is the basic identity information you will need as you proceed into analytic geometry and calculus. It is not really difficult if we take the trouble to lay our groundwork. It is all progressive.
It is time to lay out our bibliography for these study notes before I go too far. I will do that next time. It is essential for two reasons. First, I will be able to acknowledge my sources. I have borrowed from many other people for these studies. Also, I want to offer the bibliography as a guide to additional study guide for any who are interested. After that, I will take up analytic geometry. Please don’t say yippee unless you mean it.