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Solve the equation tan θ = 0.7 for 180° ≤ θ ≤ 360° and simplify 52x²y²/5ab² ÷ 13x²y³/10ab².

Mathematics

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2 parts
b

θ≈215.0∘\theta \approx 215.0^\circ

c

8a2+b2b28a^{2+b^2}b^2

Step-by-step solution

10 steps

Part b· 4 steps

Solve the equation tan⁡θ=0.7\tan \theta = 0.7 for 180∘≤θ≤360∘180^\circ \le \theta \le 360^\circ.

  1. Find the basic angle α\alpha such that tan⁡α=0.7\tan \alpha = 0.7.

    α=tan⁡−1(0.7)\alpha = \tan^{-1}(0.7) α≈34.992∘\alpha \approx 34.992^\circ

  2. Determine the quadrants where tan⁡θ\tan \theta is positive.

    The tangent function is positive in Quadrant I and Quadrant III.

  3. Identify the solution within the given range 180∘≤θ≤360∘180^\circ \le \theta \le 360^\circ.

    • A solution in Quadrant I (θ=α\theta = \alpha) is 34.992∘34.992^\circ, which is not within the given range.
    • A solution in Quadrant III is θ=180∘+α\theta = 180^\circ + \alpha. This falls within the range.

  4. Calculate the value of θ\theta in Quadrant III.

    θ=180∘+34.992∘\theta = 180^\circ + 34.992^\circ θ=214.992∘\theta = 214.992^\circ

    Rounding to one decimal place:

    θ≈215.0∘\boxed{\theta \approx 215.0^\circ}

Part c

8a2+b2b28a^{2+b^2}b^2

Simplify 52x2y35a−b2÷13x2y310a2b2\frac{52x^2y^3}{5a^{-b^2}} \div \frac{13x^2y^3}{10a^2b^2}.

  1. Rewrite the division as multiplication by the reciprocal of the second fraction.

    52x2y35a−b2×10a2b213x2y3\frac{52x^2y^3}{5a^{-b^2}} \times \frac{10a^2b^2}{13x^2y^3}

  2. Combine the numerators and denominators.

    52×10×x2×y3×a2×b25×13×x2×y3×a−b2\frac{52 \times 10 \times x^2 \times y^3 \times a^2 \times b^2}{5 \times 13 \times x^2 \times y^3 \times a^{-b^2}}

  3. Cancel out common terms from the numerator and denominator.

    The terms x2x^2 and y3y^3 appear in both the numerator and denominator, so they cancel.

    52×10×a2×b25×13×a−b2\frac{52 \times 10 \times a^2 \times b^2}{5 \times 13 \times a^{-b^2}}

  4. Simplify the numerical coefficients.

    5213=4\frac{52}{13} = 4 105=2\frac{10}{5} = 2

    Multiply these simplified numerical values: 4×2=84 \times 2 = 8.

  5. Simplify the terms involving aa and bb using exponent rules.

    For the aa terms: a2a−b2=a2−(−b2)=a2+b2\frac{a^2}{a^{-b^2}} = a^{2 - (-b^2)} = a^{2+b^2}.
    The b2b^2 term remains in the numerator.

  6. Combine all simplified parts.

    8×a2+b2×b28 \times a^{2+b^2} \times b^2 8a2+b2b2\boxed{8a^{2+b^2}b^2}
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