Given tanA=34, find the other trigonometric ratios of the angle A.
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Step-by-Step Solution
Step 1: Define the given tangent ratio
We are given the value of tanA. In a right-angled triangle, the tangent of an angle is defined as the ratio of the length of the side opposite to the angle to the length of the side adjacent to the angle. We can represent this with a right triangle △ABC, where ∠B is the right angle.
Step 2: Assign side lengths using a constant
From the ratio ABBC=34, we can assume the lengths of the sides BC and AB are 4k and 3k respectively, where k is a positive constant. This maintains the given ratio while allowing for scaling of the triangle.
Step 3: Calculate the hypotenuse using Pythagoras Theorem
Now we use the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This allows us to find the length of the hypotenuse, AC.
Step 4: Calculate sinA
The sine of angle A is the ratio of the length of the side opposite to angle A to the length of the hypotenuse. We substitute the values we found for BC and AC.
Step 5: Calculate cosA
The cosine of angle A is the ratio of the length of the side adjacent to angle A to the length of the hypotenuse. We substitute the values we found for AB and AC.
Step 6: Calculate cosec A
The cosecant of angle A is the reciprocal of sinA. We use the value of sinA calculated in the previous step.
Step 7: Calculate secA
The secant of angle A is the reciprocal of cosA. We use the value of cosA calculated in the previous step.
Step 8: Calculate cotA
The cotangent of angle A is the reciprocal of tanA. We use the given value of tanA.