find the matching restatement of the expression csc 0

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travis19

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which of the following is equal to csc0

a 1/sin0
b 1/cos0
c 1/tan0
b 1/sec0
 


If you tell me the definition of cosecant, then I will tell you the answer to this exercise.

 
Re: csc 0

travis19 said:
which of the following is equal to csc0

a 1/sin0
b 1/cos0
c 1/tan0
b 1/sec0
Click to expand...

To type ? with standard PC keyboard, type <ALT>233 - on the number pad with num-lock-key on. DO NOT type 0 to represent ?.
 
csc ?- ratio of the hypotenuse to the opposite side of a right-angled triangle
 
travis19 said:
csc ?- ratio of the hypotenuse to the opposite side of a right-angled triangle
Click to expand...

and what is the definition of sin(?) and cos(?)?

Do you see a relationship csc (?) between sin(?) or cos(?)?
 


You gave the definition in terms of right-triangle trigonometry, so you're talking about angles whose terminal side lies in Quadrant I, when the angle is placed in standard position on a rectangular coordinate system (i.e., the vertex of the right triangle is at the origin).

I suppose that's okay. (Perhaps, you have not yet learned about radian measure.)

Are you familiar with compound ratios?

In symbols: A/(B/C) = A * (C/B)

In words: Dividing some number A by some fraction B/C gives the same value as multiplying the number A by the reciprocal of B/C.

Let's make the following assignments for the numbers A, B, and C, in terms of the right-triangle you're talking about.

A = 1

B = O = length of your triangle's side Opposite angle Theta

C = H = length of your triangle's Hypotenuse

1/(O/H) = 1 * (H/O) = H/O

In other words, if we divide the fraction O/H into the number 1, we get the fraction H/O.

Can you see how this statement is equivalent to the following statement?

If we divide a Sine ratio into the number 1, we get a Cosecant ratio.

You just told me that the ratio H/O is the definition of Cosecant for angle Theta (for right-triangles drawn in Quadrant I, where the vertex with angle Theta is at the origin).

H/O = csc(?)

Well, I just showed you how the same ratio H/O is equal to the compound ratio 1/(O/H). So, we can write:

1/(O/H) = csc(?)

Now, you should know that the ratio O/H above is the same as the Sine of angle Theta, right? So, we can replace the ratio O/H above with the Sine of angle Theta.

1/sin(?) = csc(?)

There is your answer!

Again, the reciprocal identities for Cosecant, Secant, and Cotangent are, in general, not introduced in terms of right triangles.

I think it makes more sense to first introduce radian measure, so that ALL six trigonometric definitions are viewed as FUNCTIONS of Real numbers. This way, the definitions and identities hold for ALL angles, positive and negative, large and small, representing more than one revolution or less than one revolution, regardless of whether triangles are involved or not.

I do not know what you've been taught, so far. The following definitions hold for ANY* Real number ?.

csc(?) = 1/sin(?)

sec(?) = 1/cos(?)

cot(?) = 1/tan(?)

tan(?) = sin(?)/cos(?)

cot(?) = cos(?)/sin(?)

*It should go without saying that none of these ratios are defined for any value of Theta that causes the denominator to equal zero.

I REPEAT, this is much easier to understand after you've been introduced to the concept of radians -- especially when using a unit circle, where radians are equal to a distance traveled around the circumference of the unit circle (in either direction).

Then, we are not limited to right-triangles drawn in Quadrant I involving an angle at the origin that is restricted to something between 0 and 90 degrees.

Cosecant is the reciprocal of Sine.

Secant is the reciprocal of Cosine.

Cotangent is the reciprocal of Tangent.

 
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