Simplify the following expressions
a) cos(180° + x) + cos(180° - x)
b) cos(126°)cos(36°) + sin(126°)sin(36°)
can you please show me how to do this
You must have been studying the identity:
cos(A+B) = cosAcosB - sinAsinB
cos(A-B) = cosAcosB + sinAsinB
so in a) apply both of those, and knowing that
cos 180° = -1 and sin 180° = 0
simplify
for b)
compare cos(126°)cos(36°) + sin(126°)sin(36°)
with cos(A-B) = cosAcosB + sinAsinB
let me know what you got
Sure! Let's simplify the given expressions step by step.
a) For cos(180° + x) + cos(180° - x):
Step 1: Recall the periodicity of the cosine function. The cosine function has a period of 360°, which means that cos(x + 360°) = cos(x) for any angle x.
Step 2: Use the sum and difference identities for cosine. The sum identity states that cos(A + B) = cos(A)cos(B) - sin(A)sin(B), and the difference identity states that cos(A - B) = cos(A)cos(B) + sin(A)sin(B).
Step 3: Let's apply these identities to simplify the expression:
cos(180° + x) + cos(180° - x)
= cos(180°)cos(x) - sin(180°)sin(x) + cos(180°)cos(x) + sin(180°)sin(x)
= (-1)(cos(x)) - 0(sin(x)) + (-1)(cos(x)) + 0(sin(x))
= -cos(x) - cos(x)
= -2cos(x)
Therefore, the simplified expression is -2cos(x).
b) For cos(126°)cos(36°) + sin(126°)sin(36°):
Step 1: Recall the angle addition identity for sine and cosine. The angle addition identities state that sin(A + B) = sin(A)cos(B) + cos(A)sin(B), and cos(A + B) = cos(A)cos(B) - sin(A)sin(B).
Step 2: Let's apply these identities to simplify the expression:
cos(126°)cos(36°) + sin(126°)sin(36°)
= cos(90° + 36°)cos(36°) + sin(90° + 36°)sin(36°)
= [cos(90°)cos(36°) - sin(90°)sin(36°)]cos(36°) + [sin(90°)cos(36°) + cos(90°)sin(36°)]sin(36°)
= [0(cos(36°)) - 1(sin(36°))]cos(36°) + [1(cos(36°)) + 0(sin(36°))]sin(36°)
= -sin(36°)cos(36°) + cos(36°)sin(36°)
= 0
Therefore, the simplified expression is 0.
I hope this explanation helps you understand how to simplify these expressions!