How do you solve for secant with given cotangent and cosecant?

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In summary: Thanks for the help everyone!In summary, the student is trying to solve a problem for which they have not been formally taught. They start by using trig identities and substitution to solve for sin and cos, and are confident that the answer is correct. However, they ask for help with understanding what the answer is, and are told that it is the square root of a trig identity. They are also told that it can be positive or negative.
  • #1
e^(i Pi)+1=0
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Homework Statement


Given that cot [itex]\theta[/itex] = -12/5 and csc [itex]\theta[/itex] < 0, find sec[itex]\theta[/itex].

This was a question on a test that I drew a blank on, and I'm still not sure how to handle it due to my "teacher" repeatedly dismissing me when I try asking about it. Now, it occurred to me that this could be solved using trig identities and substitution. Starting with [itex]sin^{2}[/itex][itex]\theta[/itex] + [itex]cos^{2}[/itex][itex]\theta[/itex] = 1 and sin/cos = -5/12, I ended up with -
sin = -5/13
cos= 12/13
tan= -5/12
sec= 13/12

and I am confident this is the right answer. But, we have not covered trig identities in class so I am sure there is another easier way to solve this. My question is...what is it? Also, say you take the square root of a trig identity in an equation - how do you know weather it is positive or negative? Thanks in advance.
 
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  • #2
hi e^(i Pi)+1=0! :smile:
e^(i Pi)+1=0 said:
… we have not covered trig identities in class so I am sure there is another easier way to solve this. My question is...what is it?

that is the way :smile:

(though it would be easier to memorise and use sec2 = tan2 + 1, csc2 = cot2 + 1 :wink:)

(another way of course is to say that if cot = 12/5, then it's obviously a 5,12,13 triangle, and then eg sec will be hyp/adj)
Also, say you take the square root of a trig identity in an equation - how do you know weather it is positive or negative?

you need to be told (as in this question)

btw, i can't see any latex :redface: … are other people having this problem?​
 
  • #3
e^(i Pi)+1=0 said:

Homework Statement


Given that cot [itex]\theta[/itex] = -12/5 and csc [itex]\theta[/itex] < 0, find sec[itex]\theta[/itex].

This was a question on a test that I drew a blank on, and I'm still not sure how to handle it due to my "teacher" repeatedly dismissing me when I try asking about it. Now, it occurred to me that this could be solved using trig identities and substitution. Starting with [itex]sin^{2}[/itex][itex]\theta[/itex] + [itex]cos^{2}[/itex][itex]\theta[/itex] = 1 and sin/cos = -12/5, I ended up with -
sin = -12/13
cos= 5/13
tan= -5/12
csc= -13/12

and I am confident this is the right answer. But, we have not covered trig identities in class so I am sure there is another easier way to solve this. My question is...what is it? Also, say you take the square root of a trig identity in an equation - how do you know weather it is positive or negative? Thanks in advance.

Even if you haven't covered [itex]sin^{2}[/itex][itex]\theta[/itex] + [itex]cos^{2}[/itex][itex]\theta[/itex] = 1 formally, I guess you could envisage a right-angled triangle with adjacent 12 and opposite 5, and get the hypotenuse with Pythagoras, for one.
 
  • #4
Sometimes I don't see latex, but it always pops up after I refresh. Thank you for the quick responses.

edit - actually, my answer WAS wrong since I started with tan = -12/5 when it was -5/12, but it's fixed now.
 
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1. What is trigonometry and why is it important?

Trigonometry is a branch of mathematics that deals with the study of triangles and their relationships. It is important because it is used in a variety of fields such as engineering, physics, and navigation, and is essential for understanding concepts like angles, distance, and direction.

2. What are the basic trigonometric functions?

The basic trigonometric functions are sine, cosine, and tangent, which are commonly denoted as sin, cos, and tan respectively. These functions represent the ratios of the sides of a right triangle, and are used to solve various mathematical problems involving triangles.

3. How do you find the value of a trigonometric function?

The value of a trigonometric function can be found by using a calculator or by looking up the values on a trigonometric table. It can also be calculated manually using the Pythagorean theorem and the known values of the sides of a right triangle.

4. What is the unit circle and how is it related to trigonometry?

The unit circle is a circle with a radius of 1 unit, centered at the origin on a Cartesian plane. It is used in trigonometry to visualize the values of the trigonometric functions for different angles. The x-coordinate of a point on the unit circle is the cosine of the corresponding angle, while the y-coordinate is the sine.

5. How can I apply trigonometry in real-life situations?

Trigonometry is used in various real-life situations such as calculating the height of a building, determining the distance between two points, or even in video games to create realistic 3D environments. It is also used in fields like astronomy, architecture, and surveying to make accurate measurements and calculations.

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