Question

can anyone guide me how to solve questions like of domain and range of trigonometric functions . like for eg. finde the range of sinX+cosX . explain with this type of example​

Answer 1

Answer:

range of given function. We can write sinx + cosx as 2^(1/2 )sin(x+45). Since the range of sine function is[-1,1], so the range of this function will be [-(2^(1/2)),2^(1/2)]...for uques

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Step-by-step explanation:

Answer 2

Answer:

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Step-by-step explanation:

Therefore,

the domain of y = sinxsin⁡x and y = cosxcos⁡x is the set of all real numbers

the range is the interval [-1, 1], or -1 ≤ y ≤ 1.

Cosec x or cscxcsc⁡x

We know that, cscxcsc⁡x = \( \frac{1}{\sin {x}} \). Therefore,

the domain of y = cscxcsc⁡x is the set {x: x ∈ R and x ≠ nπ, n ∈ Z}

the range is the set {y: y ∈ R, y ≥ 1 or y ≤ -1}

secxsec⁡x

We know that, secxsec⁡x = 1cosx1cos⁡x. Therefore,

the domain of y = secxsec⁡x is the set {x: x ∈ R and x ≠ (2n + 1)π2π2, n ∈ Z }

the range is the set {y: y ∈ R and y ≤ -1 or y ≥ 1}

tanxtan⁡x

We know that, tanxtan⁡x = sinxcosxsin⁡xcos⁡x. Therefore,

the domain of y = tanxtan⁡x is the set {x: x ∈ R and x ≠ (2n + 1)π2π2, n ∈ Z }

the range is the set of all real numbers

cotxcot⁡x

We know that, cotxcot⁡x = cosxsinxcos⁡xsin⁡x. Therefore,

the domain of y = cotxcot⁡x is the set {x: x ∈ R and x ≠ nπ, n ∈ Z}

the range is the set of all real numbers.

The following table describes the behavior of these trigonometric functions in all four quadrants where x increases from 0 to π2π2, π2π2 to π, π to 3π23π2, and 3π23π2 to 2π.

Quadrant IQuadrant IIQuadrant IIIQuadrant IVsinincreases from 0 → 1decreases from 1 → 0decreases from 0 → -1increases from -1 → 0cosdecreases from 1 → 0decreases from 0 → -1increases from -1 → 0increases from 0 → 1tanincreases from 0 → ∞increases from -∞ → 0increases from 0 → ∞increases from -∞ → 0cotdecreases from ∞ → 0decreases from 0 → -∞decreases from ∞ → 0decreases from 0 → -∞secincreases from 1 → ∞increases from -∞ → -1decreases from -1 → -∞decreases from ∞ → 1cosecdecreases from ∞ → 1increases from 1 → ∞increases from -∞ → -1decreases from -1 → -∞

Graphical Representations of Trigonometric Functions

We already know that the values of sinxsin⁡x and cosxcos⁡x repeat after an interval of 2π. This can be shown as follows:

Hence, the values of secxsec⁡x and cscxcsc⁡x will also repeat after an interval of 2π. This can be shown as follows:

However, the values of tanxtan⁡x repeat after an interval of π. Also, the values of cotxcot⁡x which is the inverse of tanxtan⁡x will repeat after an interval of π. This can be shown as follows:

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