Question

If \(1^\circ \approx 0.01745\) radians, then the approximate value of \[ \sec 29^\circ \] is:

• A. \(1.1530\)
• B. \(1.1430\)
• C. \(1.1525\)
• D. \(1.1493\)

Hint:

For small values of \(h\) (in radians), \[ \cos(\theta-h)\approx \cos\theta+h\sin\theta. \] This approximation is frequently used in objective examinations to estimate trigonometric values near standard angles.

Answer 1

The Correct Option is B

Concept: When an angle is close to \(30^\circ\), the value of a trigonometric function can be approximated using differential corrections or small-angle approximations. Since \[ 29^\circ = 30^\circ-1^\circ, \] and \[ 1^\circ \approx 0.01745 \text{ radians}, \] we use the linear approximation \[ \cos(\theta-h)\approx \cos\theta+h\sin\theta. \]

Step 1: Express \(29^\circ\) as \(30^\circ-1^\circ\). \[ \cos29^\circ = \cos(30^\circ-1^\circ). \] Using \[ \cos(\theta-h) \approx \cos\theta+h\sin\theta, \] where \[ h=1^\circ=0.01745. \]

Step 2: Substitute known values. We know \[ \cos30^\circ=\frac{\sqrt3}{2}\approx0.8660, \] and \[ \sin30^\circ=\frac12. \] Therefore \[ \cos29^\circ \approx 0.8660+(0.01745)\left(\frac12\right). \] \[ \cos29^\circ \approx 0.8660+0.008725. \] \[ \cos29^\circ \approx 0.874725. \]

Step 3: Calculate \(\sec29^\circ\). Since \[ \sec29^\circ = \frac1{\cos29^\circ}, \] we get \[ \sec29^\circ \approx \frac1{0.874725}. \] \[ \sec29^\circ \approx 1.1432. \]

Step 4: Match with the options. The nearest option is \[ \boxed{1.1430}. \] Hence the correct answer is \[ \boxed{\text{(B)}\ 1.1430}. \]