COMPOUND ANGLES
1) Find the values of
a) sin75°.
b) cos105°.
c) sin15°.
d) cos15°.
e) cot15°.
2) Show that: tan(A - B) tan(A - B)= (sin²A - sin²B)/(cos²A - sin²B).
3) show that: cot(A+ B) cot(A - B)= (cos²A - sin²B)/(sin²A - sin²B).
4) show that cos75° = (√3-1)/2√2.
5) show: cos80°40'. cos39°20' - sin80°40' . sin39°20' = -1/2.
6) Prove: cot2A + tanA = cosec2A.
7) If tan θ= (a sinx + b sin y)/(a cos x + b cos y) show that a sin( θ- x)+ b sin( θ- y)= 0.
8) If cosA = 3/5 and sinB= 5/13 (A, B < 90°) show that cos(A - B)= 56/65.
9) show that: cos²x + cos(60° - x)+ cos²(60°+ x)= 3/2.
10) Show that sin²A + sin²(120°- A)+ sin²(120°+ A)= 3/2.
11) Show that cos²(A - 129°)+ cos²A + cos²(A + 120°)= 3/2.
12) Find the quadrant in which the angle (x + y) terminate if sin x = 3/5; cos y = -5/13, x being in the first and y being in the second quadrant. sin(x + y) is positive and cos(x + y) is negative, x + y is in the second quadrant
13) Express cos θ+ √3 sin θ in the form r cos(x - θ) and determine the value of r and x.
14) Find the maximum and minimum values of 5 cos θ+ 12 sin θ + 12.
15) If A+ B = 225°, show that tanA + tanB = 1- tanA tanB.
16) If A + B + C=90°, show that tanB tanC + tanC tanA + tanA tanB = 1.
17) An angle θ is divided into two parts x and y, such that tan x : tan y = K: 1, show that sin(x - y)= {(K -1) sin θ}/(K +1).
18) If sinx sin y - cos x cos y +1= 0, show that 1+ cot x tan y = 0.
19) If m tan( θ- 30°)= n tan( θ- 120°), show that cos2θ = (m + n)/2(m - n).
20) Show that {cos x + cos(y + z)}{cos x + cos(y - z)}= cos²x + cos²y + cos²z + 2 cosx cos y cos z -1.
21) If cos(x - y)= -1, show that cosx + cos y = 0 and sin x + sin y = 0 (x, y have real valus).
22) If cos(b - y) + cos(y - a) + cos(a - b)= -3/2, then show that cos a + cos b + cos y = 0 and sin a + sin b sin y = 0.
23) Let 0< x < π, 0< y <π and cos x + cos y - cos(x + y) = 3/2. Show that x= y =π/3.
24) Given that tan x= 3/4, cos y= -12/13 and x and y are in the same quadrant and lie between 0° and 360°. Find without use of tables the value of
a) sin(x - y).
b) cos(x/y).
25) If sin(x + y)= 4/5, cos(x - y)= 12/13 and 0< x<π/4, 0< y < π/4, find tan2x.
26) If x and y are the solutions of a tan θ+ b sec θ= c show that tan(x + y)= 2ac/(a²- c²).
27) If x and y are two distinct values of θ (0≤ x<2π, 0≤ y ≤2π) satisfying the equation sin( θ+x)= (1/2) sin2x. Show that (sin x + sin y)/+cos x + cos y)= cot x.
TRIGONOMETRICAL RATIOS OF ALLIED ANGLES
1) The value of:
a) sin(-30°).
b) cos(-200°).
c) sin210°.
d) tan(-150°).
e) sec150°.
f) cot240°.
g) sin(-225°).
h) cos390°.
i) cot315°.
j) sec510°.
k) sin(-300°).
l) sin480°.
m) tan(1860°).
n) sin315°.
o) tan(-570°).
p) cos330°.
2) Find the value of sin135° cos315° + sin420° cos330°.
3) If tanθ = -5/11, find the values of sin θ and cosθ.
4) For any quadrilateral ABCD, show that cos(1/2) (A+ B)+ cos(1//2)(C + D)= 0.
5) If sinθ + cosθ= 0 and θ lies in the second quadrant, find the value of θ, cosecθ, and cotθ.
6) Find the value of θ between 0° and 360° if sin²θ= 3/4.
7) If 0°<θ <360°, find the value of θ in the equation cotθ + tanθ = 2 secθ.
8) Prove that tan(π/12) tan(5π/12)tan(7π/12) tan(11π/12)= 1.
9) Prove that sin²(π/8) + sin²(3π/8)+ sin²(5π/8) + sin²(7π/8)= 2.
10) Prove that the value of
{sin³(2π- x) cos³(2π- x)}/{cos²(3π/2 + x) sin³(2π + x)} . {tan(π- x)/{cosec²(π - x)}. sec²(π+ x)}/sinx is independent of x.
11) 3[sin⁴(3π/2 - x) + sin⁴(3π+ x)]- 2[sin⁶(π/2 + x)+ sin⁶(5π - x)] is independent of x.
12) If tan25°= a, show that {(tan155° - tan 115°)/(1+ tan155°. Tan115°)}= (1- a²)/2a.
13) If sec(x - y)= 1, sec(x + y)= 2/√3, find positive magnitude of x and y.
14) Find the value of this sin{nπ + (-1)ⁿ π/3} when n is any positive integer.
15) Show that sin θ+ sin(π+ θ)+ sin(2π+ θ)+ sin(3π+ θ)+ .....to n terms is either 0 or sin θ according to n is even or odd.
TRIGONOMETRIC RATIOS OF SOME STANDARD ANGLES
1) Verify: sin60° = (2tan30°)/(1+ tan30°).
2) Show that (4/3) cot²30°+ 3 sin²60° - 2 cosec²60° - (3/4) tan²30° = 10/3.
3) Find the value of: {sin(π/3)+ sin(π/6)}/{cos(π/3)+ cos(π/6)} + {sin(π/3) - sin(π/6)}/{cos(π/3) - cos(π/6)}.
4) If α and β positive acute angles such that sin(2α - β)= 1/√2 and tan(α +β)= 1. Find α and β.
5) For a triangle ABC, the angle A is obtuse and sin(B + C)= √3/2 and tan(B - C)= 1/√3. Find the angles of the triangle.
6) Solve for θ if 0≤θ ≤ 90°.
a) cosθ + √3 sinθ = 2. 60°
b) cotθ + tanθ = 2 secθ. 30°
21/10/25
TRIGONOMETRICAL RATIOS AND IDENTITIES
1) If cosθ = 12/13, find other t-ratios of θ.
2) Prove that for real values of x, sinθ= x + 1/x is an impossible equation.
3) Eliminate θ from the given equilation x = r cosθ and y = r sinθ. x²+ y²= r²
4) If cosecθ+ cotθ = 2+ √3, find the values of sinθ and cosθ. 1/2, √3/2
5) Prove:
a) (3- 4 sin²θ)/cos²θ= 3- tan²θ.
b) sinθ/(cotθ + cosecθ)= 2+ sinθ/(cotθ - cosecθ).
c) (tanθ + secθ -1)/(tanθ - secθ +1)= (1+ sinθ)/cosθ.
d) sec²x tan²y - tan²x sec²y = tan²y - tan²x.
e) sinθ(1+ tanθ)+ cosθ(1+ cotθ)= secθ+ cosecθ.
f) √{(1+ sinx)/(1- sinx)} - secx = secx - √{(1- sinx)/(1+ sinx)}.
g) If cosθ - sinθ= √2 sinθ show cosθ + sinθ= √2 cosθ.
h) If cosθ - sinθ= √2 sinθ show cosθ + sinθ= ± √2 cosθ.
i) If tan²x = 1+ 2 tan²y; show that cos²y = 2 cos²x.
j) Find tanx if 2 sin²x - 5 sinx cosx + 7 cos²x = 1.
k) If k tanθ= tan kθ, show that (sin²kθ)/sin²θ= = k²{1+ (k²-1) sin²θ}.
l) If a= b cosC + c cosB = c cosA + a cosC and c= a cosB + b cosA, prove that a/sinA = b/sinB = c/sinC.
m) If x= a secθ cosφ, y= b secθ sinφ and z= c tanθ, show that x²/a²+ y²/b²- z²/c²= 1.
n) If cosθ + secθ= √3, show that cos³θ+ sec³θ= 0.
o) If m= cosecA - sinA= secA - cosA then show that tanA = (n/m)¹⁾³.
p) show (1- sinA+ cosA)²= 2(1- sinA)(1+ cosA).
q) If (sinx - cosx)/(sinx + cosx)= tanθ, show that sinx + cosx =√2 cosθ.
r) If tanθ + sinθ= m and tanθ - sinθ= n, show that m²- n²= 4√(mn).
s) If sinθ+ cosecθ= 2 show that sinⁿθ + coseⁿθ= 2.
t) If tanθ= a/b, find the value of sinθ/cos⁸θ+ cosθ/sin⁸θ. (a⁹+ b⁹)(a²+ b²)⁷⁾²/(ab)⁸
u) If secx sec y + tan x tan y = sec z show that sec x tan y + tan x sec y = ± tan z.
v) If (sin⁴θ)/a + (cos⁴θ)/b = 1/(a + b), show that (sin⁸θ)/a³ + (cos⁸θ)/b³= 1/(a+ b)³.
LIMITS
1) lim ₓ→₂ [(x²-4)/{√(3x -2) - √(x +2)}].
2) lim ₓ→₀ [{√(2- x) - √(2+ x)}/x].
3) lim ₓ→₀{(xᵐ - aᵐ)/(xⁿ - aⁿ)}.
4) lim ₓ→₃{(x⁵- 243)/(x²-9)}.
5) lim ₓ→ₐ {√(x+2)³ - √(a +2)³}/(x - a).
6) lim ₓ→₂ {xⁿ - 2ⁿ)/(x -2)}= 80 and n ∈N, find n.
7) lim ₓ→₁ {(1- x⁻¹⁾³)/(1- x⁻²⁾³)}.
8) lim ₓ→₁ {(x⁴-1)/(x -1)}= lim ₓ→ₖ{(x³- k³)/(x²- k²)}. Find the value of k.
9) lim ₓ→₀ {(1+ x)⁶ -1 }/{(1+ x)² -1}.
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