please someone teach me how to work with this types of problem.

Question

1) use the giving values to evaluate the other four trigonometric functions.

         csc(theta)=53/45 , cot =28/45

 

 

2) Match the trigonometric expression

tan theta(cot theta +tan theta)

 

  With the one of the following 

a)csc theta

 b)sin theta 

 c) sin^2 theta

d)sin theta tan theta 

 e)sec^2 theta

f) sec^2 theta +tan^2 theta

   

   I think is f

 

Please someone explan i always ask this types of questions about trigonometry i really still dont understand

Answer 1

Daisy,

 

I have a basic illustration of a right triangle with the hypotenuse, opposite and adjacent sides labeled relative to a specified theta which you can cut an paste if you want a more visual picture.  http://www.wyzant.com/resources/files/264737/basic_trigonometric_definition_illustration

 

The basic definitions are

sin θ = opp/hyp

cos θ = adj/hyp

tan θ = opp/adj

 

And the inverse definitions are

csc θ = 1 / sin θ = hyp/opp

sec θ = 1 / cos θ = hyp/adj

cot θ = 1 / tan θ = adj/opp

 

For the first question, you are given that the csc θ = 53/45; and the cot θ = 28/45 and you need to find the sin, cos, tan and sec (the other four relationships).

 

The first two are straightforward.  Since the csc θ is the reciprocal of the sin θ, the sin θ has to be the reciprocal of the csc θ.  So if the csc θ = 53/45, then the sin θ = 45/53.  The same reciprocal relationship exists between the cot and tan (which I'll leave for you to do).

 

The cos is a little more difficult.  Here another relationship between the sin, cos and tan is useful. 

opp/adj = tan θ = sin θ / cos θ = (opp/hyp)/(adj/hyp)   If you write the sin/cos in terms of the adjacent, opposite and hypotenuse, you see that the hypotenuses will cancel out and leave only the opposite over the adjacent, the definition of the tan.

 

So manipulating, this equation by multiplying both sides by cos and dividing both sides by tan gives

cos θ = sin θ / tan θ

So all you need to do is divide your answer for the sin above, by your answer for the tan above.

Once you have the cos, the sec has the same reciprocal relationship to the cos that you used to find the sin and tan.

2)  So I think you are trying to manipulate the following equation.

(tan θ)(cot θ + tan θ)

 

Generally, you want to find (and eliminate) any of the 3 pythagorean identities and put everything into the same kind of trigonometric terms.  Usually, I recommend converting everything into sin and cos (using the tan=sin/cos and cot=cos/sin), but in this case, you can convert everything into tangents, so you should rewrite cot and 1/tan

 

(tan θ)(1/(tan θ) + tan θ)

 

Then you need to distribute the tan across both terms or

 

(tan θ / tan θ) + (tan θ)(tan θ) which equals 1 + tan²θ which is one of the 3 identities, isn't it?  (f) is not the right answer.  I hope this helps.  John

 

 

Answer 2

/*** Trig in a Nutshell

Graph a general point P(x,y) in Quadrant I
(works in any Quadrant; just more intuitive in I).

Draw a segment from A(0,0) to P(x,y).
Draw a segment from A(0,0) to B(x,0).
Draw a segment from B(x,0) to P(x,y).
Label the segment lengths: AB = x, BP = y, AP = r.

By the Pythagorean Theorem r = √(x^2+y^2),
where r is defined as positive.

Label the angle BAP as θ.

Since P(x,y) := P( r cos(θ), r sin(θ) ),
where ":=" means "is defined as", then:

sin(θ) = y/r
cos(θ) = x/r
tan(θ) := sin(θ)/cos(θ) = y/x
cot(θ) := 1/tan(θ) = x/y
sec(θ) := 1/cos(θ) = r/x
csc(θ) := 1/sin(θ) = r/y

***/

 

Let's use this Nutshell to solve our problems:

 

"1) use the giving values to evaluate the other four trigonometric functions."

 

cot(θ) =28/45 = x/y

csc(θ)=53/45 = r/y

 

Which gives us: x = 28, y = 45, and r = 53.

 

Just substitute into these:

 

sin(θ) = y/r = 45/53
cos(θ) = x/r = 28/53
tan(θ) := sin(θ)/cos(θ) = y/x = 45/28
cot(θ) := 1/tan(θ) = x/y = 28/45
sec(θ) := 1/cos(θ) = r/x = 53/28
csc(θ) := 1/sin(θ) = r/y = 53/45

"2) Match the trigonometric expression
tan(θ) (cot(θ) + tan(θ))

With the one of the following
a) csc(θ)
b) sin(θ)
c) sin^2(θ)
d) sin(θ)tan(θ)
e) sec^2(θ)
f) sec^2(θ) +tan^2(θ)

I think is f "

 

tan(θ) (cot(θ) + tan(θ)) =

(y/x)((x/y) + (y/x)) =

1 + (y/x)^2 =

1 + tan^2(θ); but that's not one of the choices, so ...

1 + (y/x)^2 =

(x/x)^2 + (y/x)^2 =

(x^2 + y^2)/x^2 =

r^2/x^2 = (r/x)^2 = sec^2(θ) ==> answer is e.

Answer 3

CSC Θ = 53/45

 

SIN θ = 45/53

 

COS Θ =√((1 - (45/53)^2) = 0.53

TAN θ = ( 45/53) / 0.53 = 1.60

 

COTΘ = 1/1.60 = 0.625

 

SECθ = 1/COS Θ = 1/0.53 = 1.89

 

2)

 

 TANΘ(COTθ + TANΘ )  =

 

SINΘ .  *   (COS θ + SINθ)

COSΘ       SINΘ    COSθ

 

SINΘ   * COS²Θ + SIN² Θ

COSΘ    SIN θ COSΘ

 

  SINΘ   *         1              =    1         = SEC²θ

COSθ        SIN Θ COSΘ         COS²Θ

 

e) is the correct answer.