Trigonometric Identities

Question

If cos(θ)=24/25, and θ is in Quadrant I, then what is cos(θ/2)?

Answer 1

Cos(x + x) = cos(x)cos(x) – sin(x)sin(x) = cos² (x) – sin²(x)

Cos(2x) = cos² (x) – sin²(x)

Cos(x) = cos² (x/2) – sin²(x/2)

cos² (x/2) + sin²(x/2) =1

sin²(x/2) =1 - cos² (x/2)

Cos(x)= cos² (x/2) – [1- cos² (x/2)]

Cos(x) = cos² (x/2) –1 + cos² (x/2)

Cos(x) = 2cos² (x/2) –1

Cos(x) +1 = 2cos² (x/2)

cos² (x/2) = (1/2)[cox(x) +1

cos (x/2) = sqr{(1/2)[cox(x) +1}                 (1)

Sin (x) = 24/25   

                            Cos (x) = 7/25                 (2)

Substitute form (2) into (1)

cos (x/2) = sqr{(1/2)[7/25 +1}= sqr{(1/2)[32/25}=

sqr{16/25} =+ (4/5) or – (4/5)

Cos is Positive in Quadrant I, Cos (x/2) = (4/5) 

Final answer is Cos (x/2) = (4/5) 

Answer 2

cos(x/2) = 0.99 x=theta

cos(x/2) = sqr(1+cosx)/2 use the half angle formula

cos(x/2) = sqr(1+24/25)/2 = sqr(49/25)/2

=+ or - 7/[5 sqr(1/2)]

= 7/5(2)^1/2 = + or 7/7.07 = = 0.99 or almost 1

Alternatively cosx = adjacent side/hypotenuse = 24/25. that angle is close to zero

cos^-1(24/25) = 16.26 degrees or -16.26 degrees; use a calculator with inverse cosine function, then take half

cos(16.26/2) = cos(8.13) = 0.99

The cosine of very small angles is nearly 1. 24/25 is 0.96 and close to one

Cut the angle in half, and it gets even closer to 1, 0.99 is very close to one.

Quadrant I or II both have cosine the same for + very small angle or - very small angle.