What is the value of (sin 15 + cos 105)? (a) 0 (b) 2sin 15 (c) cos 15 + sin 15 (d) sin 15 – cos 15
Hint: To solve this expression given by (sin 15 + cos 105), we will use the property that cos(90+x) = -sinx. We will then substitute this trigonometric property into the above expression with x = 15 to get the required answer.
Complete step by step answer: Before we solve this problem, we should be aware of a few basics of solving trigonometric expressions. We generally try to express all the degrees of the trigonometric expressions in the same value (if possible). For example, if we want to compare sin 20 and cos 70, we try to convert one of them so that we get the same value of the degree. Thus, in this case, we use the trigonometric property that cos(90-x) = sinx. In this case, cos 70 = cos (90 – 20) = sin 20. Thus, we are finally able to tell that sin 20 and cos 70 are equal. Now, coming on to the problem in hand, we have to solve (sin 15 + cos 105). We can write – cos 105 = cos (90+15) Now, we use the property that cos(90+x) = - sinx, thus, we get, = - sin 15 Finally, we get, = sin 15 + cos 105 = sin 15 - sin 15 = 0 Hence, the correct answer is (a) 0.
Note: Although, we don’t have tanx, cotx, secx and cosecx in this problem, we try to express these trigonometric expressions in terms of sinx and cosx. This would help us in evaluating the expression with ease. Further, it is also important to be aware about other related properties like sin(90+x) = cosx, cos(180-x) = -cosx, etc (these are related results to the problem in hand and may come in handy while solving related problems).