问题1.评估以下内容：
(i)正弦60°cos 30°+正弦30°cos 60°
解决方案：
Formulas to be used : sin 30° = 1/2
cos 30° = √3/2
sin 60° = 3/2
cos 60° = 1/2
=> (√3/2) * (√3/2) + (1/2) * (1/2)
=> 3/4 +1/4
=> 4 /4
=> 1
(ii)2 tan 2 45°+ cos 2 30°– sin 2 60°
解决方案：
Formulas to be used : sin 60° = √3/2
cos 30° = √3/2
tan 45° = 1
=> 2(1)(1) + (√3/2)(√3/2)-(√3/2)(√3/2)
=> 2 + 3/4 – 3/4
=> 2
(iii)cos 45°/(秒30°+ cosec 30°)
解决方案：
Formulas to be used : cos 45° = 1/√2
sec 30° = 2/√3
cosec 30° = 2
=> 1/√2 / (2/√3 + 2)
=> 1/√2 / (2+2√3)/√3
=> √3/√2×(2+2 √3) = √3/(2√2+2√6)
=> √3(2√6-2√2)/(2√6+2√2)(2√6-2√2)
=> 2√3(√6-√2) / (2√6)²-(2√2)²
=> 2√3(√6-√2)/(24-8) = 2 √3(√6-√2)/16
=> √3(√6-√2)/8
=> (√18-√6)/8
=> (3√2-√6)/8
(iv)(正弦30°+棕褐色45°–毫秒60°)/(秒30°+ cos 60°+轻便床45°)
解决方案：
Formulas to be used : sin 30° = 1/2
tan 45° = 1
cosec 60° = 2/√3
sec 30° = 2/√3
cos 60° = 1/2
cot 45° = 1
=> (1/2+1-2/√3) / (2/√3+1/2+1)
=> (3/2-2/√3)/(3/2+2/√3)
=> (3√3-4/2 √3)/(3√3+4/2 √3)
=> (3√3-4)(3√3-4)/(3√3+4)(3√3-4)
=> (27+16-24√3) / (27-16)
=> (43-24√3)/11
(ⅴ)(5cos 2 60°+ 4秒2 30° -褐色2 45°)/(2罪30°+cos²30°)
解决方案：
Formulas to be used : cos 60° = 1/2
sec 30° = 2/√3
tan 45° = 1
sin 30° = 1/2
cos 30° = √3/2
=> 5(1/2)2+4(2/√3)²-1²/(1/2)+(√3/2)
=> (5/4+16/3-1) / (1/4+3/4)
=> (15+64-12) / 12/(4/4)
=> 67/12
问题2.选择正确的选项并证明您的选择合理：
(i)2tan 30°/ 1 + tan 2 30°=
(A)正弦60°(B)余弦60°(C)棕褐色60°(D)正弦30°
(ii)1-tan 2 45°/ 1 + tan 2 45°=
(A)棕褐色90°(B)1(C)正弦45°(D)0
(iii)sin 2A = 2当A =
(A)0°(B)30°(C)45°(D)60°
(iv)2tan30°/ 1 tan 2 30°=
(A)cos 60°(B)正弦60°(C)棕褐色60°(D)正弦30°
解决方案：
(i) In the given equation, substituting the value of tan 30°
As tan 30° = 1/√3
2tan 30°/1+tan230° = 2(1/√3)/1+(1/√3)2
=> (2/√3)/(1+1/3) = (2/√3)/(4/3)
=> 6/4√3 = √3/2
=> sin 60°
The ans is sin 60°.
The correct option is (A).
(ii) In the given equation, substituting the of tan 45°
As tan 45° = 1
1-tan245°/1+tan245° = (1-12)/(1+12)
= 0/2 => 0
The ans is 0.
The correct option is (D).
(iii) sin 2A = 2 sin A is true when A = 0°
sin 2A = sin 0° = 0
2 sin A = 2 sin 0° = 2 × 0 = 0
Another way :
sin 2A = 2sin A cos A
=> 2sin A cos A = 2 sin A
=> 2cos A = 2 => cos A = 1
Now, we have to check which degree value has to be applied, to get the solution as 1.
When 0 degree is applied to cos value we get 1, i.e., cos 0 = 1
Hence, A = 0°
The correct option is (A).
(iv) As tan 30° = 1/√3
2tan30°/1-tan230° = 2(1/√3)/1-(1/√3)2
=> (2/√3)/(1-1/3) = (2/√3)/(2/3) = √3 = tan 60°
The correct option is (C).
问题3.如果tan(A + B)=√3且tan(A – B)= 1 /√3，则0° B，找到A和B。
解决方案：
tan (A + B) = √3
tan (A + B) = tan 60°
(A + B) = 60° … (i)
tan (A – B) = 1/√3
tan (A – B) = tan 30°
(A – B) = 30° … (ii)
Now add the equation (i) and (ii), we get
A + B + A – B = 60° + 30°
A= 45°
Substituting the value of A in equation (i) to find the value of B
45° + B = 60°
B = 60° – 45°
B = 15°
Hence, A = 45° and B = 15°
问题4.陈述以下是对还是假。证明你的答案。
(i)罪(A + B)=罪A +罪B.
(ii)sinθ的值随θ的增加而增加。
(iii)cosθ的值随θ的增加而增加。
(iv)对于所有θ值，sinθ= cosθ。
(v)没有为A = 0°定义轻便小床A。
解决方案：
(i) Let us take A = 60° and B = 30°, then
Substitute the values of A and B in the sin (A + B) formula, we get
sin (A + B) = sin (60° + 30°) = sin 90° = 1 and,
sin A + sin B = sin 60° + sin 30°
= √3/2 + 1/2 = (√3 + 1 ) / 2, sin(A + B) ≠ sin A + sin B
Since both the values obtained are not equal.
Hence, the statement is false.
(ii) From the values given below, we can see that as angle(theta) increases value also increases.
sin 0° = 0, sin 30° = 1/2, sin 45° = 1/√2, sin 60° = √3/2 , sin 90° = 1
Thus, the value of sin θ increases as θ increases.
Hence, the statement is true.
(iii) From the values given below, we can see that as angle (theta) increases value decreases.
cos 0° = 1, cos 30° = √3/2 , cos 45° = 1/√2, cos 60° = 1/2, cos 90° = 0
Thus, the value of cos θ decreases as θ increases.
Hence, the statement given above is false.
(iv) sin θ = cos θ, is only true for theta = 45°
Therefore, the above statement is false.
(v) As tan 0° = 0
cot 0° = 1 / tan 0°
= 1 / 0 => undefined
Hence, the given statement is true.