Trigonometry Chapter 3~ Properties of Trigonometric Functions
Example 1 (3.1) Prove that 1+tan²x = sec²x
1+ sin²x/ cos²x= sec²x , cos²x/cos²x + sin²x/sin²x= sec²x , cos²x +sin²x/ cos²x = sec² x , 1/cos²x= sec²x . Sec²x= sec²x
Example 1 (3.2) Prove 1+t²/1-t² - t/1-t = 1/1+t
1+t²/(1+t)(1-t) - t/1-t =1/1+t => 1+t²-t-t²/(1+t)(1-t)=1/1+t =>1-t/(1+t)(1-t) = 1/1+t=> 1/1+t=1/1+t
Example 3 (3.1) Express 1 - sin x cos x tan x in terms of cos x
1-sin x *cos x * sin x/ cos x = 1-sin²x = cos²x
Strategies for Proving Identities
1. If one side of a trig. identity is a sum or difference of two rational expressions, try combing over a common denominator. 2. Try converting one side of the identity into an expression in terms of just sin x and/or cos x. 3. If one side of the identity is given in factored form, try multiplying the parts of the factored expression. 4. If one side of the identity can be factored,try factoring that side. Keep in mind:a²-b²= (a+b)(a-b) or a²+2ab+b²= (a+b)² or a²-2ab+b²= (a-b)² . 5. Work with the more complicated side of the equation first. 6. The presence of squared functions (i:e.:sin²x) should suggest the use of one of the Pythagorean Identities. 7. Remember that you can use the alternative forms of certain identities(because of sin²x+cos²x=1, you can use sin²x=1-cos²x). 8. At last resort,try simplifying both sides of the equation to the same expression.
Example 2 (3.2) Prove sec x-cos x= sin x*tan x
1/cos x- cos x/1= sin x*tan x, 1/cos x- cos²x/cos x= sin x *tan x, 1-cos²x/cos x= sin x *tan x, sin²x/cos x = sin*sin x/cos x => sin²x/cos x = sin²x/cos x
Example 2 (3.1) Express csc²x + cot²x in terms of sin x
1/sin²x + cos²x/ sin²x , 1+cos²x/ sin²x => 1+ (1-sin²x)/sin²x => 2-sin²x/ sin²x
Difference Identity for cos
Cos(a-b)= cos a*cos b+sin a*sin b
Difference Identity for sin
Sin (a-b)= sin a*cos b - cos a*sin b
Trigonometric identities
Tan x= sin x/cos x and cot x= cos x/sin x
Double Angle Identity cos
cos (2x) = cos²x- sin²x or cos(2x)=2cos²x-1 or cos(2x)= 1- 2sin²x
Sum identity for cos
cos (a+b)= cos a*cos b- sin a *sin b
Half angle identity cos
cos (x/2) =±√1+cos x/2
Double Angle Identity sin
sin (2x) = 2 sin x cos x
Sum Identity for sin
sin (a+b)=sin a * cos b+cos a* sin b
Half angle identity sin
sin (x/2) =±√1-cos x/2
Reciprocal Identities
sin x = 1/csc x , csc x = 1/sin x , cos x = 1/sec x , sec x = 1/cos x , tan x = 1/cot x , cot x = 1/tan x
Example 3 (3.2) sin x/csc x + cos x/sec x= sec²x-tan²x
sin x/1+sin x +cos x/1/cos x = sec²x-tan²x => sin²x+cos²x= 1+tan²x-tan²x , 1=1
Pythagorean Identities
sin²x + cos²x = 1 , 1+tan²x= sec²x , 1+cot²x=csc²x , (sin²x= 1-cos²x) , (tan²x= sec²x-1) , (cot²x= csc²x-1) , (cos²x= 1-sin²x) , (sec²x- tan²x=1) , (csc²x - cot²x =1)
Double Angle Identity for tan
tan (2x)= 2 tan x/ 1-tan²x
Sum identity for tan
tan (a+b) = tan a+tan b/1-tan a*tan b
Difference identity for tan
tan (a-b)= tan a-tan b/ 1+tan a *tan b
Half Angle Identity for tan
tan (x/2) = sin x/ 1+cos x or tan (x/2)= 1-cos x/sin x or tan(x/2)= ±√1-cos x/1+cos x