Trigonometric Problems
Exercise 1: Prove that:
A) , if
, if α
I quadrant,
, β
I quadrant.
B) , if
, if α
I quadrant,
, β
I quadrant.
C) , if
,
and
, where α, β, γ
I quadrant.
Exercise 2: Prove that if , than
A) tgα.tgβ + tgβ.tgγ + tgα.tgγ = 1.
B) ctgα + ctgβ + ctgβ = ctgα.ctgβ.ctgγ.
Exercise 3: Prove that if
then (1 + tgα)(1 + tgβ) = 2.
Exercise 4: Prove that
A) cosα + √3sinα ≤ 2.
B) √2(sinα + cosα) ≤ 2.
C) sin2αcosβ + cos2αsinβ ≤ 1.
D) sin(α + β) ≤ sinα + sinβ.
E) cos(α + β) < cosα - sinα.sinβ.
F) √3cos&lpha; - sinα ≤ 2.
G) sinα - cosα ≤ √2.
Exercise 5: Find the maximal value of y = √3sin2x - cos2x and the value of x, if 0 ≤ x ≤ π.
Exercise 6: Calculate the examples
A) sin67°.cos68° - cos67°.sin68°.
B) sin27°.cos33° + sin63°.cos57° + cos30°.
C)
D)
E) cos1° + cos121° + cos241°.
F)
Exercise 7: Prove the equations
A)
B) cosα - √3sinα = 2cos(60° + α).
C) tg20° + tg23° + tg20°.tg25° = 1.
D) tg20° + tg40° + √3tg20°.tg40° = √3.
E) sin(α + β).cos(α - β) = sin2α - sin2β.
F) cos(α + β).cos(α - β) = cos2α - cos2β.
G)