Question
QUESTION 4
4.1
Simplify, without using a calculator:
4.1.1
𝑠𝑖𝑛 (−135°)·𝑠𝑖𝑛 240°·𝑐𝑜𝑠 25°·𝑡𝑎𝑛 480° 𝑠𝑖𝑛 115°·𝑐𝑜𝑠 135°
(7)
4.1.2
(2𝑠𝑖𝑛(270°+𝑥)+𝑐𝑜𝑠(𝑥−180°))× 𝑠𝑖𝑛𝑥 2𝑐𝑜𝑠 (90°+𝑥)+𝑠𝑖𝑛 (180°+𝑥) 𝑐𝑜𝑠𝑥
(6)
4.1.3
𝑐𝑜𝑠 120° 3( 𝑠𝑖𝑛 45° ) 4 .2 𝑐𝑜𝑠45°
(4)
4.2
Prove the identity: 1−𝑠𝑖𝑛 θ + 1−𝑠𝑖𝑛 θ = 2 1+𝑠𝑖𝑛 θ 1−𝑠𝑖𝑛 θ 1+𝑠𝑖𝑛θ
(5)
4.3
Determine the general solution of 𝑡𝑎𝑛 2θ =− 2
(5)
4.1
Simplify, without using a calculator:
4.1.1
𝑠𝑖𝑛 (−135°)·𝑠𝑖𝑛 240°·𝑐𝑜𝑠 25°·𝑡𝑎𝑛 480° 𝑠𝑖𝑛 115°·𝑐𝑜𝑠 135°
(7)
4.1.2
(2𝑠𝑖𝑛(270°+𝑥)+𝑐𝑜𝑠(𝑥−180°))× 𝑠𝑖𝑛𝑥 2𝑐𝑜𝑠 (90°+𝑥)+𝑠𝑖𝑛 (180°+𝑥) 𝑐𝑜𝑠𝑥
(6)
4.1.3
𝑐𝑜𝑠 120° 3( 𝑠𝑖𝑛 45° ) 4 .2 𝑐𝑜𝑠45°
(4)
4.2
Prove the identity: 1−𝑠𝑖𝑛 θ + 1−𝑠𝑖𝑛 θ = 2 1+𝑠𝑖𝑛 θ 1−𝑠𝑖𝑛 θ 1+𝑠𝑖𝑛θ
(5)
4.3
Determine the general solution of 𝑡𝑎𝑛 2θ =− 2
(5)
Answers
Answered by
GPT 3.5
4.1.1
Starting with the first part of the expression:
sin(-135°) = sin(180° - 135°) = sin(45°) = 1/√2
Then, continuing with the rest of the expression:
sin 240° = sin(240° - 180°) = sin 60° = √3/2
cos 25° = cos(90° - 25°) = sin 65° = 1/√2
tan 480° = tan(480° - 360°) = tan 120° = √3
Putting it all together: (1/√2)(√3/2)(1/√2)(√3)(cos 115°)(cos 135°)
= (1/√2)(√3/2)(1/√2)(√3)(-1/√2)(-1/√2)
= (1/2)(√3/2)(1/2)(√3)(1/2)(1/2)
= 3/16
So, the simplified value of the expression is 3/16.
4.1.2
For this expression, expand and simplify each part separately, then combine them all together:
(2sin(270° + x) + cos(x - 180°)) * sin(2x) + cos(90° + x) * sin(180° + x) / cos(x)
= [2sin 270°cosx + 2cos270°sinx + cosx - cos180°sinx] * sin2x + [sin90°cosx + cos90°sinx] * sin180°cosx / cosx
= [0 + 2(-sinx) + cosx + 0] * sin2x + [cosx] * 0 / cosx
= (-2sinx + cosx) * sin2x
= -2sinxsin2x + cosxsin2x
= -2sin2xsinx + sin2xcosx
= -sin4x + 0
= -sin4x
Therefore, the simplified expression is -sin4x.
4.1.3
Calculate each part separately and then simplify:
cos120° = -1/2
sin45° = 1/√2
cos45° = 1/√2
So, the expression becomes:
(-1/2)(3(1/√2))(4) / (1/√2)
= -6√2
Therefore, the simplified value of the expression is -6√2.
4.2
To prove the identity:
1 - sinθ + 1 - sinθ = 2 / (1 + sinθ)(1 - sinθ)
First, simplify the left side:
1 - sinθ + 1 - sinθ = 2 - 2sinθ
Now, rewrite the right side to find a common denominator:
(2) / ((1 + sinθ)(1 - sinθ)) = 2 / (1 - sin^2θ) = 2 / cos^2θ
Since sin^2θ + cos^2θ = 1, we can rewrite this as:
2 / (1 - sin^2θ) = 2 / cos^2θ
Therefore, the left side equals the right side, proving the identity.
4.3
To determine the general solution of tan2θ = -2, we first solve for θ:
tan2θ = -2
tan2θ = tan(-63.43°) or tan(116.57°)
So, the general solutions are:
2θ = -63.43° + n180° or 116.57° + n180°
θ = (-63.43°/2) + n90° or (116.57°/2) + n90°
θ = -31.715° + n90° or 58.285° + n90°
Therefore, the general solution of tan2θ = -2 is θ = -31.715° + n90° or 58.285° + n90°, where n is an integer.
Starting with the first part of the expression:
sin(-135°) = sin(180° - 135°) = sin(45°) = 1/√2
Then, continuing with the rest of the expression:
sin 240° = sin(240° - 180°) = sin 60° = √3/2
cos 25° = cos(90° - 25°) = sin 65° = 1/√2
tan 480° = tan(480° - 360°) = tan 120° = √3
Putting it all together: (1/√2)(√3/2)(1/√2)(√3)(cos 115°)(cos 135°)
= (1/√2)(√3/2)(1/√2)(√3)(-1/√2)(-1/√2)
= (1/2)(√3/2)(1/2)(√3)(1/2)(1/2)
= 3/16
So, the simplified value of the expression is 3/16.
4.1.2
For this expression, expand and simplify each part separately, then combine them all together:
(2sin(270° + x) + cos(x - 180°)) * sin(2x) + cos(90° + x) * sin(180° + x) / cos(x)
= [2sin 270°cosx + 2cos270°sinx + cosx - cos180°sinx] * sin2x + [sin90°cosx + cos90°sinx] * sin180°cosx / cosx
= [0 + 2(-sinx) + cosx + 0] * sin2x + [cosx] * 0 / cosx
= (-2sinx + cosx) * sin2x
= -2sinxsin2x + cosxsin2x
= -2sin2xsinx + sin2xcosx
= -sin4x + 0
= -sin4x
Therefore, the simplified expression is -sin4x.
4.1.3
Calculate each part separately and then simplify:
cos120° = -1/2
sin45° = 1/√2
cos45° = 1/√2
So, the expression becomes:
(-1/2)(3(1/√2))(4) / (1/√2)
= -6√2
Therefore, the simplified value of the expression is -6√2.
4.2
To prove the identity:
1 - sinθ + 1 - sinθ = 2 / (1 + sinθ)(1 - sinθ)
First, simplify the left side:
1 - sinθ + 1 - sinθ = 2 - 2sinθ
Now, rewrite the right side to find a common denominator:
(2) / ((1 + sinθ)(1 - sinθ)) = 2 / (1 - sin^2θ) = 2 / cos^2θ
Since sin^2θ + cos^2θ = 1, we can rewrite this as:
2 / (1 - sin^2θ) = 2 / cos^2θ
Therefore, the left side equals the right side, proving the identity.
4.3
To determine the general solution of tan2θ = -2, we first solve for θ:
tan2θ = -2
tan2θ = tan(-63.43°) or tan(116.57°)
So, the general solutions are:
2θ = -63.43° + n180° or 116.57° + n180°
θ = (-63.43°/2) + n90° or (116.57°/2) + n90°
θ = -31.715° + n90° or 58.285° + n90°
Therefore, the general solution of tan2θ = -2 is θ = -31.715° + n90° or 58.285° + n90°, where n is an integer.