If sin A=3/5 and A is an acute angle, find the values of cos A and tan A.
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Step-by-Step Solution
Step 1: Use the trigonometric identity
We are given the value of sinA and we need to find cosA. The fundamental trigonometric identity sin2A+cos2A=1 relates these two quantities, so we will use it to find the value of cosA.
Step 2: Substitute the value of sin A
Substitute the given value sinA=53 into the identity. Squaring 53 gives 259.
Step 3: Simplify and solve for cos A
Subtract 259 from both sides to isolate cos2A. Then, find a common denominator and subtract the fractions. Finally, take the square root of both sides. Since A is an acute angle (between 0∘ and 90∘), cosA must be positive.
Step 4: Calculate tan A
Now that we have both sinA and cosA, we can find tanA using its definition as the ratio of sinA to cosA. This is another fundamental trigonometric identity.
Step 5: Substitute values of sin A and cos A
Substitute the given value of sinA=53 and the calculated value of cosA=54 into the formula for tanA. Simplify the complex fraction by multiplying by the reciprocal of the denominator.