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Given Tan A = (3)/(8).

Mathematics

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7 steps

  1. Identify the given information and define the sides of a right-angled triangle.

    Given Tan A=38\text{Tan A} = \frac{3}{8}.
    In a right-angled triangle, Tan A=OppositeAdjacent\text{Tan A} = \frac{\text{Opposite}}{\text{Adjacent}}.
    Let the Opposite side be 3x3x and the Adjacent side be 8x8x for some constant xx.

  2. Calculate the Hypotenuse using the Pythagorean theorem.

    Hypotenuse2=Opposite2+Adjacent2\text{Hypotenuse}^2 = \text{Opposite}^2 + \text{Adjacent}^2 Hypotenuse2=(3x)2+(8x)2\text{Hypotenuse}^2 = (3x)^2 + (8x)^2 Hypotenuse2=9x2+64x2\text{Hypotenuse}^2 = 9x^2 + 64x^2 Hypotenuse2=73x2\text{Hypotenuse}^2 = 73x^2 Hypotenuse=73x2=x73\text{Hypotenuse} = \sqrt{73x^2} = x\sqrt{73}

  3. Calculate Sin A\text{Sin A}.

    Sin A=OppositeHypotenuse=3xx73=373\text{Sin A} = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{3x}{x\sqrt{73}} = \frac{3}{\sqrt{73}}

    To rationalize the denominator:

    Sin A=373×7373=37373\text{Sin A} = \frac{3}{\sqrt{73}} \times \frac{\sqrt{73}}{\sqrt{73}} = \frac{3\sqrt{73}}{73}

    The value of Sin A\text{Sin A} is 37373\boxed{\frac{3\sqrt{73}}{73}}.

  4. Calculate Cos A\text{Cos A}.

    Cos A=AdjacentHypotenuse=8xx73=873\text{Cos A} = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{8x}{x\sqrt{73}} = \frac{8}{\sqrt{73}}

    To rationalize the denominator:

    Cos A=873×7373=87373\text{Cos A} = \frac{8}{\sqrt{73}} \times \frac{\sqrt{73}}{\sqrt{73}} = \frac{8\sqrt{73}}{73}

    The value of Cos A\text{Cos A} is 87373\boxed{\frac{8\sqrt{73}}{73}}.

  5. Calculate Sec A\text{Sec A}.

    Sec A\text{Sec A} is the reciprocal of Cos A\text{Cos A}.

    Sec A=1Cos A=1873=738\text{Sec A} = \frac{1}{\text{Cos A}} = \frac{1}{\frac{8}{\sqrt{73}}} = \frac{\sqrt{73}}{8}

    The value of Sec A\text{Sec A} is 738\boxed{\frac{\sqrt{73}}{8}}.

  6. Calculate Cosec A\text{Cosec A}.

    Cosec A\text{Cosec A} is the reciprocal of Sin A\text{Sin A}.

    Cosec A=1Sin A=1373=733\text{Cosec A} = \frac{1}{\text{Sin A}} = \frac{1}{\frac{3}{\sqrt{73}}} = \frac{\sqrt{73}}{3}

    The value of Cosec A\text{Cosec A} is 733\boxed{\frac{\sqrt{73}}{3}}.

  7. Calculate Cot A\text{Cot A}.

    Cot A\text{Cot A} is the reciprocal of Tan A\text{Tan A}.

    Cot A=1Tan A=138=83\text{Cot A} = \frac{1}{\text{Tan A}} = \frac{1}{\frac{3}{8}} = \frac{8}{3}

    The value of Cot A\text{Cot A} is 83\boxed{\frac{8}{3}}.

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