If [tex]$\cos A=\frac{3}{5}$[/tex] and [tex]$A$[/tex] is acute, find [tex]$\cot A$[/tex].
Answer (1)
Given cos A = 5 3 , we find sin A using the Pythagorean identity. Then, we calculate cot A using the formula cot A = s i n A c o s A .
Use the Pythagorean identity to find sin A : sin 2 A + ( 5 3 ) 2 = 1 , which gives sin A = 5 4 .
Apply the definition of cotangent: cot A = s i n A c o s A .
Substitute the values of cos A and sin A : cot A = 5 4 5 3 .
Simplify to find the final answer: 4 3 .
Explanation
State the given information and objective. We are given that cos A = 5 3 and that A is an acute angle. We want to find cot A .
Use the Pythagorean identity to find sin A. Since cos A = 5 3 , we can use the Pythagorean identity sin 2 A + cos 2 A = 1 to find sin A . Substituting the given value of cos A into the identity, we have sin 2 A + ( 5 3 ) 2 = 1
Solve for sin A. Solving for sin A , we get sin 2 A = 1 − 25 9 = 25 16 Since A is an acute angle, sin A is positive. Therefore, sin A = 25 16 = 5 4
Calculate cot A. Now we can find cot A using the definition cot A = s i n A c o s A . Substituting the values of cos A and sin A , we have cot A = 5 4 5 3 = 5 3 ⋅ 4 5 = 4 3 Thus, cot A = 4 3 .
Examples
Understanding trigonometric ratios like cotangent is crucial in fields like surveying and navigation. For instance, surveyors use angles of elevation and depression, along with trigonometric functions, to determine distances and heights of land features. If a surveyor knows the cosine of an angle and needs to find the cotangent to calculate a specific distance, this problem demonstrates the necessary steps.
