Trigonometric Identities



Summary of Trigonometric Identities:
sin2 x + cos2 x = 1
tan2 x + 1 = sec2 x
cot2 x + 1 = cosec2 x
Compound Angle
Formulas:		sin (A + B) = sin A cos B + cos A sin B
sin (A - B) = sin A cos B - cos A sin B
cos (A + B) = cos A cos B - sin 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)
tan (A - B) = (tan A - tan B) / (1 + tan A tan B)
Double Angle
Formulas:		sin 2x = 2 sin x cos x
cos 2x = cos2 x - sin2 x = 2 cos2 x - 1 = 1 - 2 sin2 x
tan 2x = 2 tan x / (1 - tan2 x)
t Formulas:sin x = 2t / (1 + t2)
cos x = (1 - t2) / (1 + t2)
tan x = 2t / (1 - t2)		 where t = tan (x/2)
R Formulas:a sin x + b cos x = R sin (x + alpha)
a sin x - b cos x = R sin (x - alpha)
a cos x - b sin x = R cos (x + alpha)
a cos x + b sin x = R cos (x - alpha)		 where R = sqrt(a2 + b2)
alpha = tan-1 (b/a)
Factor
Formulas:		sin A + sin B = 2 sin ((A + B)/2) cos ((A - B)/2)
sin A - sin B = 2 cos ((A + B)/2) sin ((A - B)/2)
cos A + cos B = 2 cos ((A + B)/2) cos ((A - B)/2)
cos A - cos B = -2 sin ((A + B)/2) sin ((A - B)/2)		 to change sums/differences
into products
Factor
Formulas:		sin A cos B = 1/2 [ sin (A + B) + sin (A - B) ]
cos A sin B = 1/2 [ sin (A + B) - sin (A - B) ]
cos A cos B = 1/2 [ cos (A + B) + cos (A - B) ]
sin A sin B = -1/2 [ cos (A + B) - cos (A - B) ]		 to change products
into sums/differences


Exercises for students:

Express 3 sin x + 4 cos x in the form R sin (x + a), where R is positive and a is acute, giving the value of a to the nearest 0.1°.


Use the fact that 15° = 45° - 30°, find the value of sin 15° in surd form, simplifying your answer.


Find an expression for tan 4x in terms of tan x.


Express 5 cos x - 12 sin x in the form R cos (x + a), where R > 0 and a is acute.


Express (sin 5x + sin 3x)/2 in the form a sin bx cos cx, giving the values of the constants a, b and c.


Show that, for all x, cos 4x = 8 cos4 x - 8 cos2 x + 1.


Prove that cos x - cos 3x = 4 sin2 x cos x.


Express 5 cos x + 6 sin x in the form R cos (x - a), where R > 0 and a is acute.




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