plz plz help. /trigonometry?

Question

Note that the point value of each question is given in parentheses


3. (10) Using exact values , find:
a. tan 105°
b. sin 255°


4. (5) Find cos (x+y) when both x and y are acute angles and sin x = 3/5 and sin y= 5/13.

Answer 1

3.

a) tan 105° = tan (60° + 45°) = (tan 60° + tan 45°)/(1 - tan 60° tan 45°)

tan 60° = √3, tan 45° = 1, so

tan 105° = (√3 + 1)(1 - √3(1)) × (1 + √3)/(1 + √3) = (4 + 2√3)/-2 = -2 - √3

b) sin 255° = sin(210° + 45°) = sin 210° cos 45° - cos 210° sin 45°

sin 45° = cos 45° = √2/2

sin 210° = -sin 30° = -1/2, cos 210° = -cos 30° = -√3/2

sin 255° = (-1/2)(√2/2) + (-√3/2)(√2/2) = -√2/4 - √6/4 = -(√2 + √6)/4

4) I assume since they say that both angles are acute that they are both in the 1st quadrant.

sin x = 3/5 means cos x = 4/5 (3-4-5 triangle)
sin x = 5/13 means cos x = 12/13 (5-12-13 triangle)

cos(x + y) = cos x cos y - sin x sin y = (4/5)(12/13) - (3/5)(5/13) = 48/65 - 15/65 = 33/65

Answer 2

4. Find cos (x+y) when both x and y are acute angles and sin x = 3/5 and sin y= 5/13.

Use the relationship sin² θ + cos² θ = 1 and solve for cos θ, given sin θ.

If sin x = 3/5 then cos x = 4/5.
If sin y = 5/13 then cos y = 12/13.

Now apply the formula for angle addition for cosine.

cos(x + y) = (cos x)(cos y) - (sin x)(sin y)
= (4/5)(12/13) - (3/5)(5/13) = (48 - 15)/(5*13) = 33/65

Answer 3

ANS2: cosec^2 x - sec^2 x =(cot² x+a million)-(tan² x+a million) =a million/tan² x - tan² x ANS3: a) sin² B - cos² B = sin² B - (a million - sin² B) =2sin² B - a million b) (a million – cos² y + sin² y)² +4 sin² y cos² y = (a million - 2cos² y + 2sin² y - 2sin² y cos² y + cos^4 y + sin^4 y) +4 sin² y cos² y = a million - 2cos² y + 2sin² y + 2sin² y cos² y + cos^4 y + sin^4 y = a million - 2cos² y + 2sin² y + (sin² y + cos²y)² = 2 - 2cos² y + 2sin² y = 4sin² y c) tan 0 = 0 sec 0 = a million for this reason tan² 0 sec² 0- sec² 0+a million d)sin 0/cosec0 + cos 0/sec0 =sin 0/a million/sin 0 + cos 0/a million/cos 0 =sin² 0 + cos² 0 =a million = 0 - a million + a million = 0 = tan 0 e)sin x tan ² x cot ^3 x = sin x cot x = sin x (cos x/sin x) = cos x f)sec² x/(sec ² x-a million) = a million/cos² xtan² x = a million/sin² x = cosec² x

Answer 4

3. a. -3.732
b. -.966