CPhill

2.

a.  sin(theta) sec(theta)  =   sin(theta) *  [1/cos(theta) ]   =  sin (theta) / cos (theta)  =

tan (theta)

b. 

The cotangent  of the angle would be   adj / opp   =  d / h

So we have

cot (theta)  =  d / h

cot  (60°)  ≈  .58

.58   =  3008 / h       rearrange as

h  =  3008 / .58   ≈   5186 ft

1) 

a. Remember  that  the cosine is the reciprocal of the secant

So  

A / D   =  sec (theta)

1 / [ A / D]  =  1/ cos (theta)           [ flip both fractions over ]

D /A  =  cos(theta)

D / A   =  cos (theta)

b.   cos (theta)  =  1 / sec(theta)

c.  tan (theta) =  y / x  =  3 / 4     so x = 4

If sin (theta)  = y/r  =   3/5  then  r  = 5

So....cos (theta)  =  x / r   =   4/5

So we have :

cos (theta)  = D /A

4/5  =  .25 cm / A      rearrange as

A   =  .25 cm. / (4/5)   =  (1/4) / (4/5)  =  (1/4) * (5/4)  =  5/16  cm

Thanks, tertre  !!

And

Angle C is opposite the longest side           Angle  C is opposite the longest side

Angle D is opposite the intermediate side   Angle  M is opposite the intermediate side

Angle E is opposite the shortest side         Angle L is oposite the shortest side 

So  Δ CDE  is similar to Δ CML

Look at this

Angle B  is opposite the longest side            Angle J is opposite the longest side

Angle C is oppositie the mid-length side       Angle K is opposite the mid-length side       

Angle D is opposite the shortest sde           Angle L is opposite the shortest side

So

 ΔBCD   is similar to  ΔJKL

√1/4 /  ⁴√1/8

Manuel....I find these easiest if we convert to exponential form, first...so we have

(1/4)^(1/2)  / ( 1/8)^(1/4)   note  that we can write

(1 / 2^2)^(1/2)  / (1 / 2^3)^(1/4)

( 1 / 2)  / ( 1/ 2^(3/4) )   ...so we have

(1/2)^1  *  (1/2)^(-3/4)  =

(1/2)^(1/4)   =  ⁴√(1/2)

First one top left....we can check these ratios

9/54  = 10/60  = 12/72   = 1/6   Similar due to SSS proportionality

So  ΔBCD  is similar  to  ΔJKL

Top  right

Check these ratios

60/156  = 65 /169 

5/13  =  5/13     Similar   due to SAS proportionality

So   ΔFGH is similar to  ΔSRQ

Bottom left

Check these ratios

40/96  = 65/156  = 70/168

5/12   = 5/12  = 5/12  Similar  due to SSS proportionality

So  ΔCDE is similar to  ΔCML

Bottom right

Not similar  -  no angles match

First one :

 ΔRQP   similar to  ΔUVW  by AA  similarity

For the second one :

Check these ratios

23/66 =  32/88  = 40/110

23/66 = 4/11 = 4/11   NOT similar

1.   (csc^2(x) - cot^2(x)) / (sin(-x)cot(x)

      Remember that  1 + cot^2 (x)  = csc^2 (x)

      So  csc^2 (x) - cot^2(x)  =  1

      Also   sin (-x)  =   - sin (x)

So  we have

1 / [ - sin (x) (cos (x)  / sin (x) ]  =

1 / (-cos (x) )  =    -  sec (x)

2.  sin ^2 (x)cos^2(x)  - cos^2(x)  =

(1  - cos^2(x) ) (cos^2 (x) ) -  cos^2(x)

cos^2(x) - cos^4 (x)   - cos^2(x)

-cos^4(x)

9)

If the little red arrows in the triangles mean "parallel," then the first two triangles are similar by AA congruency

So   Δ FGH  is simillar ΔFBC

10)  Not similar....we can see this by taking ratios of the two shortest sides....that is....

44/66   =  11/18 ???

4/6  =  11/18  ???   cross-multiply..

18 * 4  =  11 * 6   ???

72  =  66     not equal....so ....not simillar