Trigonometric Functions Of General Angles

Oscar

2007-12-30 5:37 pm

If θ takes any real values, find the range of y of the following functions.

A. y=3sin(θ/2)-4

B. y=1/(2sin^2θ+5cos^2θ)

C. y=sin^2θ+4sinθ

D. y=cos^2θ-sinθ+1

回答 (1)

嘉裕

2007-12-30 6:15 pm

✔ 最佳答案

A.
-1≤sin(θ/2)≤1
-3≤3sin(θ/2)≤3
-7≤3sin(θ/2)-4≤-1
-7≤y≤-1

B.
2sin^2θ+5cos^2θ=2+3cos^2θ
-1≤cosθ≤1
0≤cos^2θ≤1
2≤2+3cos^2θ≤5
1/2≥1/(2sin^2θ+5cos^2θ)≥1/5
1/5≤y≤1/2

C.
y=sin^2θ+4sinθ=(sin^2θ+4sinθ+4)-4=(sinθ+2)^2-4
-1≤sinθ≤1
1≤sinθ+2≤3
1≤(sinθ+2)^2≤9
-3≤(sinθ+2)^2-4≤5
-3≤y≤5

D.
cos^2θ-sinθ+1=1-sin^2θ-sinθ+1=2-(sin^2θ+sinθ)=2.25-(sinθ+0.5)^2
-1≤sinθ≤1
-0.5≤sinθ+0.5≤1.5
0≤(sinθ+0.5)^2≤2.25
0≥-(sinθ+0.5)^2≥-2.25
2.25≥2.25-(sinθ+0.5)^2≥0
0≤y≤2.25

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