Introduction / Context:
This question checks your memorisation and use of standard trigonometric ratios for special angles in degrees, here 60 degrees. Such values are widely used in school mathematics and aptitude tests, so being comfortable with them can save valuable time in exams.
Given Data / Assumptions:
- The angle is 60°, measured in degrees.
- We need to compute Tan 60° + Cosec 60°.
- Standard trigonometric values for 30°, 45°, and 60° are assumed known.
Concept / Approach:
For the special angle 60°, we recall that Tan 60° = √3 and Sin 60° = √3 / 2. Since Cosec θ is defined as 1 / Sin θ, we can find Cosec 60° from Sin 60°. Then we add the two values and simplify the expression into a single fraction with a rational denominator if needed.
Step-by-Step Solution:
Recall that Tan 60° = √3.
Also recall that Sin 60° = √3 / 2.
Therefore, Cosec 60° = 1 / Sin 60° = 1 / (√3 / 2) = 2 / √3.
Now compute Tan 60° + Cosec 60° = √3 + 2 / √3.
To add these, take a common denominator √3: √3 = 3 / √3.
So √3 + 2 / √3 = 3 / √3 + 2 / √3 = 5 / √3.
Verification / Alternative check:
You can quickly check numerically using approximate values. √3 is about 1.732, so √3 + 2 / √3 is approximately 1.732 + 2 / 1.732 which is about 1.732 + 1.155 = 2.887. The fraction 5 / √3 is also about 5 / 1.732 which is again close to 2.887. This confirms that 5 / √3 is correct.
Why Other Options Are Wrong:
Option a (5 / 3) is smaller than the approximate decimal 2.887. Option b (2 / √3) ignores the Tan 60° term. Option d (2 / 3) is far too small. Option e (√3) only accounts for Tan 60° and omits Cosec 60° completely.
Common Pitfalls:
Students sometimes confuse Tan 60° with 1 / √3, which is actually Tan 30°. Another mistake is mixing up Sin and Cosec or forgetting that Cosec θ is the reciprocal of Sin θ. Always check that you are using the correct special angle values and the right reciprocal relationships.
Final Answer:
The exact simplified value of Tan 60° + Cosec 60° is
5 / √3.
Discussion & Comments