["# How to Compute (\ an 75^\circ) Using Angle Addition Formulas", "Calculating trigonometric values like (\ an 75^\circ) can be efficiently achieved using angle addition formulas. While (75^\circ) is not a standard angle, it can be expressed as the sum of two well-known angles: (45^\circ + 30^\circ). By applying the tangent addition formula, we can compute (\ an 75^\circ) with precision and clarity.", "## Understanding the Tangent Addition Formula", "The tangent of a sum of two angles is given by:", "[
\n\ an(A + B) = \frac{\ an A + \ an B}{1 - \ an A \ an B}
\n]", "This formula is derived from sine and cosine addition identities and is a powerful tool for evaluating trigonometric functions of non-standard angles.", "## Applying the Formula to (\ an 75^\circ)", "We express (75^\circ) as:", "[
\n75^\circ = 45^\circ + 30^\circ
\n]", "Now we apply the addition formula:", "[
\n\ an 75^\circ = \ an(45^\circ + 30^\circ) = \frac{\ an 45^\circ + \ an 30^\circ}{1 - \ an 45^\circ \ an 30^\circ}
\n]", "## Substitute Known Tangent Values", "We know the exact values:", "- (\ an 45^\circ = 1)
\n- (\ an 30^\circ = \frac{1}{\sqrt{3}})", "Substitute these into the formula:", "[
\n\ an 75^\circ = \frac{1 + \frac{1}{\sqrt{3}}}{1 - 1 \cdot \frac{1}{\sqrt{3}}} = \frac{1 + \frac{1}{\sqrt{3}}}{1 - \frac{1}{\sqrt{3}}}
\n]", "## Simplify the Expression", "To simplify, multiply numerator and denominator by (\sqrt{3}) to eliminate the radicals:", "[
\n\ an 75^\circ = \frac{\sqrt{3} + 1}{\sqrt{3} - 1}
\n]", "To rationalize the denominator, multiply numerator and denominator by the conjugate (\sqrt{3} + 1):", "[
\n\ an 75^\circ = \frac{(\sqrt{3} + 1)^2}{(\sqrt{3} - 1)(\sqrt{3} + 1)} = \frac{(\sqrt{3} + 1)^2}{(\sqrt{3})^2 - (1)^2} = \frac{(\sqrt{3} + 1)^2}{3 - 1} = \frac{(\sqrt{3} + 1)^2}{2}
\n]", "Expand the numerator:", "[
\n(\sqrt{3} + 1)^2 = 3 + 2\sqrt{3} + 1 = 4 + 2\sqrt{3}
\n]", "So:", "[
\n\ an 75^\circ = \frac{4 + 2\sqrt{3}}{2} = 2 + \sqrt{3}
\n]", "## Conclusion", "Using angle addition formulas, we’ve shown that:", "[
\n\ an 75^\circ = 2 + \sqrt{3}
\n]", "This elegant result highlights the power of trigonometric identities in transforming complex angle computations into manageable expressions. Whether for solving problems in calculus, physics, or geometry, mastering such techniques enhances your mathematical toolkit efficiently and precisely.", "---", "Keywords: (\ an 75^\circ), angle addition formula, trigonometric identity, tangent addition, (\ an 45^\circ), tangent of sum, (2 + \sqrt{3})", "Meta Description: Learn how to compute (\ an 75^\circ) using the tangent addition formula with step-by-step derivation and simplification."]
![\]Question: A weather balloon is modeled as a sphere with radius $ r $ meters, and it ascends such that its volume increases at a rate proportional to $ \sin \theta $, where $ \theta $ is the angle of elevation from the ground. If the volume increase rate is given by $ \frac{dV}{dt} = k \sin \theta $, and $ \theta = 60^\circ $, compute the rate of volume increase in terms of $ k $ and $ r $.](https://soloferat.biz.id/images/question-a-weather-balloon-is-modeled-as-a-sphere-with-radius-r-meters-and-it-ascends-such-that-its-volume-increases-at-a-rate-proportional-to-sin-theta-where-theta-is-the-angle-of-elevation-fro.jpg)
