Trigonometry Trigonometric Equations 3

Question 1

Solve sin(x) + cos(x) = 1 for x in [0, 2π).

Accepted Answer

x = 0, π/2

Answer

Writing as √2 sin(x+π/4)=1 gives sin(x+π/4)=1/√2, so x+π/4=π/4 or 3π/4, yielding x=0 or π/2.

Question 2

Which method most directly solves sin(x) = cos(x)?

Accepted Answer

Divide both sides by cos(x) to obtain tan(x) = 1

Answer

Dividing both sides by cos(x) gives tan(x)=1, which is straightforward to solve without introducing extraneous solutions (provided cos(x)≠0).

Question 3

Solve cos(x) = sin(x) for x in [0, 2π).

Accepted Answer

x = π/4, 5π/4

Answer

Dividing by cos(x) gives tan(x)=1, so x=π/4 (Q1) and x=5π/4 (Q3).

Question 4

What is the period of solutions for the equation 2sin(3x) − 1 = 0?

Accepted Answer

2π/3

Answer

The function sin(3x) has period 2π/3, so the solutions repeat every 2π/3.

Question 5

Solve sin(x/2) = √2/2 for x in [0, 4π).

Accepted Answer

x = π/2, 3π/2

Answer

sin(x/2)=√2/2 gives x/2=π/4 or 3π/4 (within [0,2π)), so x=π/2 or 3π/2.

Trigonometry Practice Test