Solve "cos(15)+sin(15)" Without Calculator

[RoryP]

Homework Statement

Find the values of the following, without the use of a calculator,
cos(15)+sin(15)

Homework Equations

sin(A±B)= sinAcosB±cosAsinB
cos(A±B)= cosAcosB∓sinAsinB

The Attempt at a Solution

Without the use of a calculator i had to relate back to the 2 triangles which you get the most common values for sin, cos and tan. Namely,
sin(30)= 1/2
cos(30)= √3/2
sin(45)= 1/√2
cos(45)= 1/√2
sin(60)= √3/2
cos(60)= 1/2
sin(0)= 0
cos(0)= 1
sin(90)=1
cos(90)= 0
but i have found no use for any of these, also i didnt think tan was appropriate but I am open to correction! i tried going along the lines of gettting a trig function, with the same angle, to equal 1 so i could use the addition formulae but i had no success!
any help please guys!

 

[Mark44]

The double angle formulas for cosine can be manipulated to give half-angle formulas.
cos(2a) = cos^2(a) - sin^2(a) = 1 - 2sin^2(a) = 2cos^2(a) - 1
==> cos(a) = 1 - 2sin^2(a/2) and cos(a) = 2cos^2(a/2) - 1

Take the last two equations and solve for cos(a/2) in one and sin(a/2) in the other.

 

[Architeuthis Dux]

Great job using the common values for sin and cos to try to solve this problem without a calculator. One way to approach this problem is to use the sum-to-product formula for sin and cos, which states that sin(A)+sin(B) = 2sin((A+B)/2)cos((A-B)/2) and cos(A)+cos(B) = 2cos((A+B)/2)cos((A-B)/2). In this case, we have A=B=15, so the formula simplifies to sin(15)+cos(15) = 2sin(15)cos(0) = 2sin(15). Since we know that sin(15) is a common value, we can use the fact that sin(30) = 2sin(15)cos(15) to solve for cos(15). We can then substitute these values into the original expression to get cos(15)+sin(15) = 2sin(15)cos(15) + sin(15) = sin(30) + sin(15) = 1/2 + 1/2 = 1. Therefore, the solution to cos(15)+sin(15) is 1. Keep up the good work using problem-solving skills to solve challenging problems without the use of a calculator!