If \(\sin(\theta) = \frac{3}{5}\), and \(\theta\) is in the first quadrant, find \(\cos(\theta)\).
["How to Find (\cos(\ heta)) When (\sin(\ heta) = \frac{3}{5}) and (\ heta) is in the First Quadrant", "When dealing with trigonometric identities, one of the fundamental formulas is the Pythagorean identity:", "[\n\sin^2(\ heta) + \cos^2(\ heta) = 1\n]", "Given that (\sin(\ heta) = \frac{3}{5}) and that (\ heta) lies in the first quadrant (where all trigonometric functions are positive), our task is to find (\cos(\ heta)).", "### Step 1: Square (\sin(\ heta))", "First, square the given sine value:", "[\n\sin^2(\ heta) = \left(\frac{3}{5}\right)^2 = \frac{9}{25}\n]", "### Step 2: Substitute into the Pythagorean Identity", "Now plug this into the identity:", "[\n\frac{9}{25} + \cos^2(\ heta) = 1\n]", "### Step 3: Solve for (\cos^2(\ heta))", "Subtract (\frac{9}{25}) from both sides:", "[\n\cos^2(\ heta) = 1 - \frac{9}{25} = \frac{25}{25} - \frac{9}{25} = \frac{16}{25}\n]", "### Step 4: Take the Square Root", "[\n\cos(\ heta) = \sqrt{\frac{16}{25}} = \frac{4}{5}\n]", "Since (\ heta) is in the first quadrant, (\cos(\ heta)) must be positive, so we take the positive root.", "---", "### Conclusion", "Given that (\sin(\ heta) = \frac{3}{5}) and (\ heta) is in the first quadrant:", "[\n\cos(\ heta) = \frac{4}{5}\n]", "This value is crucial in many geometry, physics, and engineering problems where right triangles and wave functions are involved. Always remember to use the Pythagorean identity and consider the quadrant to determine the correct sign!", "---", "Keywords: (\sin(\ heta) = \frac{3}{5}), (\cos(\ heta)), first quadrant, inverse sine, trigonometry, identity, Pythagorean identity, Pythagorean theorem, (\cos(\ heta)) calculation.\nMeta description: Learn how to find (\cos(\ heta)) when (\sin(\ heta) = \frac{3}{5}) using the Pythagorean identity. Step-by-step math example for first quadrant angles."]
