State all the possible radian values of theta, to the nearest hundredth

Question

cos(θ)=-5/13 where 0≤θ≤2π

 

i got 1.97 but how do i get the other answer in quadrant 3?

 

In general, how would i find out in all 4 quadrant?

Answer 1

The cosine function has positive values in the first and fourth quadrants, and negative values in the second and third. The value you found is in the second quadrant: π/2 < 1.97 < π

 

To find the corresponding angle in the third quadrant, you first need to find the reference angle.

π - φ = 1.97

φ = π - 1.97

This is the measure of the angle between the -x-axis and the angle you've found. To put this into the third quadrant, you need to add it to the -x-axis

π + φ = the 2nd answer.

 

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In general, when you have an angle and want to find similar angles in other quadrants, you'll use the following relationships between standard angle θ and reference angle φ. All 4 of these angles will have the same absolute value of each trig function. You'll need to learn which trig functions are positive and negative in which quadrants.

 

In QI: θ = φ

QII: θ = π - φ

QIII: θ = π + φ

QIV: θ = 2π - φ,

    or θ = -φ for negative angles -π/2 < θ < 0