Trig radians: sec (2z-pi/6)=2 for 0<z <pi radians

[cired2002]

sec (2z-pi/6)=2 for 0<z <pi radians

 

[mmm4444bot]

Let's see your efforts, so far. (I know that you've been asked to show your work, at least once before.) Where did you get stuck?

My first step was to rewrite the secant function in terms of cosine.

 

[cired2002]

Let's see your efforts, so far. (I know that you've been asked to show your work, at least once before.) Where did you get stuck?

My first step was to rewrite the secant function in terms of cosine.

I expand with secant then I changed it to cosine. That is where I got stucked. I dont know how to change (1/cos pi/6) to any form.

 

[tkhunny]

I don't know how to change (1/cos pi/6) to any form.

sec(2z - pi/6) = 2

Have you considered solving

cos(2z - pi/6) = 1/2

?

 

[mmm4444bot]

I expand with secant then I changed it to cosine. That is where I got [stuck].

Please excuse me; I don't know the meaning of "expand with secant".

My thought was: switch to cosine first.

1/cos(2z - Pi/6) = 2

Next, use this pattern (to get tkhunny's equation):

^EXPRESSION / B = A

_{is the same as}

^EXPRESSION / A = B

So:

1/cos(2z - Pi/6) = 2

is the same as

1/2 = cos(2z - Pi/6)

I dont know how to change (1/cos pi/6) to any form.

Again, I don't know how you arrived at sec(Pi/6), but it seems like you have not yet learned trigonometric values for the special angles. (Pi/6 is a special angle.)

1/2 is a trigonometric value for some of the special angles. If you need to learn the special angles (between 0 and 2·Pi, inclusive), you can google keywords trig values of special angles

For example, if I see:

(√2)/2 = sin(θ)

then I know right away that (between 0 and 2·Pi)

θ = Pi/4 or θ = 3·Pi/4

because each of those are special angles (45° and 135°) for which I have memorized that (√2)/2 is the sine value.

In other words, once you know the special angles whose cosine is 1/2, you can solve equations like:

2z - Pi/6 = Angle1

2z - Pi/6 = Angle2

 

[cired2002]

Please excuse me; I don't know the meaning of "expand with secant".

My thought was: switch to cosine first.

1/cos(2z - Pi/6) = 2

Next, use this pattern (to get tkhunny's equation):

^EXPRESSION / B = A

_{is the same as}

^EXPRESSION / A = B

So:

1/cos(2z - Pi/6) = 2

is the same as

1/2 = cos(2z - Pi/6)


Again, I don't know how you arrived at sec(Pi/6), but it seems like you have not yet learned trigonometric values for the special angles. (Pi/6 is a special angle.)

1/2 is a trigonometric value for some of the special angles. If you need to learn the special angles (between 0 and 2·Pi, inclusive), you can google keywords trig values of special angles

For example, if I see:

(√2)/2 = sin(θ)

then I know right away that (between 0 and 2·Pi)

θ = Pi/4 or θ = 3·Pi/4

because each of those are special angles (45° and 135°) for which I have memorized that (√2)/2 is the sine value.

In other words, once you know the special angles whose cosine is 1/2, you can solve equations like:

2z - Pi/6 = Angle1

2z - Pi/6 = Angle2

Yeah i know the special angle and I know cos 30 = squareroot of 3/2 but I just dont know how to work with radians I.e. cos pi/6

 

[ksdhart2]

Yeah i know the special angle and I know cos 30 = squareroot of 3/2 but I just dont know how to work with radians I.e. cos pi/6

Then you know what you need to study. Learning about radians will prove very very helpful in future trigonometry exercises, as few people actually use degrees. You might try Google something like "convert radians to degrees." Here's one page I found by doing just that. The critical identity is that Pi radians = 180 degrees. And you convert between the two scales by multiplying by a form of one so the unwanted units cancel out. As an example:

$\displaystyle \dfrac{3\pi}{2} \text{ radians} \cdot \dfrac{180^{\circ}}{\pi \text{ radians}} = \dfrac{180^{\circ}}{\left(\dfrac{2}{3}\right)} = 180^{\circ} \cdot \dfrac{3}{2} = 270^{\circ}$

 

[mmm4444bot]

…I know cos 30 = squareroot of 3/2 …

First, here are some notes about notation.

When we state an angle measure without any units, it's understood to be measured in radians. If you want to say 30 degrees, then you must type a degree symbol (or the letters 'deg') after the measurement.

cos(30) means the cosine of 30 radians.

cos(30 deg) means the cosine of 30 degrees.

Also, be careful how you type ratios. The cosine of 30 degrees is not the "square root of 3/2". It is one-half the square root of 3:

cos(30 deg) = sqrt(3)/2

I'm glad that you're aware of the special angles. Next, what are the two angles whose cosine is 1/2? Those are the angles that each equal 2z-Pi/6.

… I just dont know how to work with radians

With the special angles from 0° through 180°, you're just dividing half a circle into halves, fourths, thirds, or sixths.

You know that Pi radians is 180° and that the arc subtending that angle is half the circle.

If we divide half the circle into sixths, then we have six angles each measuring 30° (180°/6=30°).

In radians, we're still dividing half the circle into sixths, but each of those six angles is 1/6th of Pi, instead of 1/6th of 180°.

 30° = 1·Pi/6 = Pi/6

 60° = 2·Pi/6 = Pi/3

 90° = 3·Pi/6 = Pi/2

120° = 4·Pi/6 = 2·Pi/3 

150° = 5·Pi/6

180° = 6·Pi/6 = Pi

Radian measure gets easier, the more you use it. Print out this chart, and refer to it often (until you don't need to, anymore). :cool: