inverse trig

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Last updated 5:38 AM on 7/25/26
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51 Terms

1
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sin-1(x) range

[−π2,π2][-\frac{\pi}{2},\frac{\pi}{2}]

2
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cos-1(x) range

[0,π][0, \pi]

3
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tan-1(x) range

(−π2,π2)(-\frac{\pi}{2}, \frac{\pi}{2})

4
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sin-1(x) domain

[−1,1][-1, 1]

5
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cos-1(x) domain

[−1,1][-1, 1]

6
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tan-1(x) domain

(−∞,∞)(-\infty, \infty)

7
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csc-1(x) range

[−π2,π2][-\frac{\pi}{2},\frac{\pi}{2}] - {0}

8
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sec-1(x) range

[0,π]−{π2,}[0,\pi]-\left\lbrace\frac{\pi}{2},\right\rbrace

9
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cot-1(x) range

[0,π]\left\lbrack0,\pi^{}\right.]

10
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csc-1(x) domain

R-(-1,1)

11
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sec-1(x) domain

R-(-1,1)

12
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cot-1(x) domain

(−∞,∞)(-\infty, \infty)

13
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sin-1(-x)

=−sin−1(x), x ∈[−1,1]=-sin^{-1}(x), \, x \, \in [-1, 1]

14
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cos-1(-x)

=π−cos−1(x), x ∈[−1,1]=\pi - cos^{-1}(x), \, x \, \in [-1, 1]

15
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tan-1(-x)

=−tan−1(x), x ∈R=-tan^{-1}(x), \, x \, \in \mathbb{R}

16
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csc-1(-x)

=−csc−1(x), ∣x∣≥1=-csc^{-1}(x), \, |x| \geq 1

17
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sec-1(-x)

=π−sec−1(x), ∣x∣≥1=\pi - sec^{-1}(x), \, |x| \geq 1

18
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cot-1(-x)

=π−cot−1(x), x ∈R=\pi - cot^{-1}(x), \, x \, \in \mathbb{R}

19
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sin-1(x) + cos-1(x)

=π2, x ∈[−1,1]=\frac{\pi}{2}, \, x \, \in [-1, 1]

20
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tan-1(x) + cot-1(x)

=π2, x ∈R=\frac{\pi}{2}, \, x \, \in \mathbb{R}

21
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sec-1(x) + csc-1(x)

=π2, ∣x∣≥1=\frac{\pi}{2}, \, |x| \geq 1

22
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sin-1(1/x)

=csc−1(x), if x≥1 or x≤−1=csc^{-1}(x), \text{ if } x \geq 1 \text{ or } x \leq -1

23
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cos-1(1/x)

=sec−1(x), if x≥1 or x≤−1=sec^{-1}(x), \text{ if } x \geq 1 \text{ or } x \leq -1

24
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tan-1(1/x)

=cot^{-1}(x), \text{ if } x > 0

25
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tan-1(x) + tan-1(y)

=tan^{-1}\left(\frac{x+y}{1-xy}\right), \text{ if the value } xy < 1

26
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tan-1(x) - tan-1(y)

=tan^{-1}\left(\frac{x-y}{1+xy}\right), \text{ if the value } xy > -1

27
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2 tan-1(x)

=sin−1(2x1+x2), ∣x∣≤1=sin^{-1}\left(\frac{2x}{1+x^2}\right), \, |x| \leq 1

28
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2tan-1(x)

=cos−1(1−x21+x2), x≥0=cos^{-1}\left(\frac{1-x^2}{1+x^2}\right), \, x \geq 0

29
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2tan-1(x)

=\tan^{-1}\left(\frac{2x}{1-x^2}\right),-1<x<1

30
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3sin-1(x)

=sin−1(3x−4x3)=sin^{-1}(3x-4x^3)

31
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3cos-1(x)

=cos−1(4x3−3x)=cos^{-1}(4x^3-3x)

32
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3tan-1(x)

=tan−1(3x−x31−3x2)=tan^{-1}\left(\frac{3x-x^3}{1-3x^2}\right)

33
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sin(sin-1(x))

=x,−1≤x≤1=x, -1 \leq x \leq 1

34
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cos(cos-1(x))

=x,−1≤x≤1=x, -1 \leq x \leq 1

35
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tan(tan-1(x))

=x, -\infty < x < \infty

36
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csc(csc-1(x))

=x, -\infty < x \leq 1 \text{ or } -1 \leq x < \infty

37
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sec(sec-1(x))

=x, -\infty < x \leq 1 \text{ or } 1 \leq x < \infty

38
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cot(cot-1(x))

=x, -\infty < x < \infty

39
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sin-1(sin(θ))

=θ,−π2≤θ≤π2=θ, -\frac{\pi}{2} \leq θ \leq \frac{\pi}{2}

40
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cos-1(cos(θ))

=θ,0≤θ≤π=θ, 0 \leq θ \leq \pi

41
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tan-1(tan(θ))

=θ, -\frac{\pi}{2} < θ < \frac{\pi}{2}

42
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csc-1(csc(θ))

=θ, -\frac{\pi}{2} \leq θ < 0 \text{ or } 0 < θ \leq \frac{\pi}{2}

43
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sec-1(sec(θ))

=θ, 0 \leq θ \leq \frac{\pi}{2} \text{ or } \frac{\pi}{2} < θ \leq \pi

44
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cot-1(cot(θ))

=θ, 0 < θ < \pi

45
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sin-1 x + sin-1 y

=sin−1(x1−y2+y1−x2), if x,y≥0 and x2+y2≤1=sin^{-1}\left(x\sqrt{1-y^2} + y\sqrt{1-x^2}\right), \text{ if } x, y \geq 0 \text{ and } x^2 + y^2 \leq 1

46
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sin-1 x + sin-1 y

=\pi-sin^{-1}\left(x\sqrt{1-y^2}+y\sqrt{1-x^2}\right),\text{ if }x,y\geq0\text{ and }x^2+y^2>1

47
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sin-1 x - sin-1 y

=sin−1(x1−y2−y1−x2), if x,y≥0 and x2+y2≤1=sin^{-1}\left(x\sqrt{1-y^2}-y\sqrt{1-x^2}\right),\text{ if }x,y\geq0\text{ and }x^2+y^2\leq1

48
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sin-1 x - sin-1 y

=\pi-sin^{-1}\left(x\sqrt{1-y^2}-y\sqrt{1-x^2}\right),\text{ if }x>0,\text{ y<0 and }x^2+y^2>1

49
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cos-1 x + cos-1 y

=cos−1(xy−1−x21−y2), if x,y>0 and x2+y2≤1=cos^{-1}\left(xy - \sqrt{1-x^2}\sqrt{1-y^2}\right), \text{ if } x, y > 0 \text{ and } x^2 + y^2 \leq 1\n\n

50
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cos-1 x + cos-1 y

=cos^{-1}\left(xy - \sqrt{1-x^2}\sqrt{1-y^2}\right), \text{ if } x, y > 0 \text{ and } x^2 + y^2 \leq 1\n\n

51
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cos-1 x - cos-1 y

=cos^{-1}\left(xy + \sqrt{1-x^2}\sqrt{1-y^2}\right), \text{ if } x, y > 0 \text{ and } x^2 + y^2 \geq 1