If sec A = \(17 \over 8\) , given that A < 90°, what is the value of the following? \(34 sin A+15 cot A \over 68 Cos A-16 tan A\)
19
Evaluating Trigonometric Expressions with Given Secant Value
We are given that sec A = \(17 \over 8\) and that A is an acute angle (A < 90°). We need to find the value of the expression \(34 sin A+15 cot A \over 68 Cos A-16 tan A\).
To evaluate the expression, we first need to find the values of sin A, cos A, tan A, and cot A.
Finding Cos A from Sec A
We know that sec A is the reciprocal of cos A.
The relationship is:
Substituting the given value of sec A:
Finding Sin A using the Pythagorean Identity
Since A is an acute angle, both sin A and cos A are positive. We can use the Pythagorean identity:
Substitute the value of cos A we found:
Now, solve for sin² A:
Since A is acute, sin A is positive. Take the square root of both sides:
Finding Tan A and Cot A
Now that we have sin A and cos A, we can find tan A and cot A.
The formula for tan A is:
Substitute the values of sin A and cos A:
The formula for cot A is the reciprocal of tan A:
Substitute the value of tan A:
Summary of Trigonometric Ratios for Angle A
Let's summarize the values we found:
| Trigonometric Ratio | Value |
|---|---|
| sin A | \(15 \over 17\) |
| cos A | \(8 \over 17\) |
| tan A | \(15 \over 8\) |
| cot A | \(8 \over 15\) |
| sec A (Given) | \(17 \over 8\) |
Evaluating the Given Expression
The expression we need to evaluate is \(34 sin A+15 cot A \over 68 Cos A-16 tan A\).
Substitute the values of sin A, cos A, tan A, and cot A into the expression.
Numerator: \(34 \text{ sin A} + 15 \text{ cot A}\)
Denominator: \(68 \text{ Cos A} - 16 \text{ tan A}\)
Now, divide the numerator by the denominator:
Final Value
The value of the expression \(34 sin A+15 cot A \over 68 Cos A-16 tan A\) is 19.
Revision Table: Key Trigonometric Identities
| Identity | Description |
|---|---|
| \( \text{sec A} = {1 \over \text{cos A}} \) | Secant is the reciprocal of Cosine. |
| \( \text{cosec A} = {1 \over \text{sin A}} \) | Cosecant is the reciprocal of Sine. |
| \( \text{cot A} = {1 \over \text{tan A}} \) | Cotangent is the reciprocal of Tangent. |
| \( \text{tan A} = {\text{sin A} \over \text{cos A}} \) | Tangent is Sine divided by Cosine. |
| \( \text{sin}^2 \text{ A} + \text{cos}^2 \text{ A} = 1 \) | The fundamental Pythagorean identity in trigonometry. |
Additional Information: Acute Angles in Trigonometry
An acute angle is an angle that measures less than 90 degrees. In trigonometry, when dealing with an acute angle A in a right-angled triangle, all trigonometric ratios (sin A, cos A, tan A, cosec A, sec A, cot A) are positive.
The fact that A < 90° is important because it ensures that:
- We take the positive square root when calculating sin A from sin² A = 1 - cos² A.
- All calculated trigonometric ratio values (sin A, cos A, tan A, cot A) are positive, which is consistent with trigonometric ratios in the first quadrant of the unit circle (where angles are between 0° and 90°).
Understanding the quadrant in which an angle lies helps determine the sign of trigonometric ratios. For acute angles (first quadrant), all ratios are positive.
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