Trigonometry #10 - Questions and Answers

Question 1 How can I prove tan 15° =2 - √3 using the triple-angle formulas? sin 3x = 3sin x - 4sin³x cos 3x = 4cos³x - 3cosx I know that if I substitute x = 15° I can write sin 45 = 3sin 15 – 4sin³15 cos 45 = 4cos³15 - 3cos 15 What can I do next?any hints Answer Let s = sin 15°, c = cos 15° and t = s/c = tan 15° ∈(tan 0°, tan 45°) = (0, 1) then, from your equations, since sin 45° = cos 45°, it follows that s(3 - 4s²) = c(4c² - 3) ⇒ s(3(s² + c²) - 4s²) = c(4c² - 3(s²+c²)) and, after dividing both sides by c³, we obtain t(3 – t²) = (1 – 3t²) ⇒ (t+1) (t² - 4t + 1) = 0 (the factorization can be obtained by noticing that the equation on the left is satisfied by t=-1). Can you take it from here and find the root t ∈ (0,1)? Question 2 What is the precise value of the expression (cos 25° - sin (-5°)) / (sin 25° + cos 5°) Answer (cos 25°-sin 5°) / (sin 25°+cos 5°) = (sin 65°-sin 5°) / (sin 25°+sin 85°) = (2sin 30°cos 35°) / (2sin 55°cos 30°) = tan 30° = 1/√3 Question 3 Prove the Identity - Double Angle cot (A) = sin(2A) / (1 - cos(2A)) Answer sin (2A) / (1 - cos (2A)) = 2sin Acos A / (1-(cos²A-sin²A)) = 2sin Acos A / (1-cos²A+sin²A) = 2sin Acos A / (sin²A+sin²A) = 2sin Acos A / 2sin²A = cos A/sin A = cot A Question 4 Trigonometry is one of the most interesting chapters of Quantitative Aptitude section. Basically, it is a part of SSC syllabus. Today I will tell you the easy method to learn all the basics of trigonometry i.e. Trigonometric Ratios, facts and formulas. Answer Trigonometric Ratios There are six trigonometric ratios. First three are the primary functions and last three are just the reciprocals of above three. Those are written as follows: (sin θ) (cos θ) (tan θ) (cot θ) (sec θ) (cosec θ) See the following image, You are already well versed with this image. This is a basic right angled triangle. Points to remember: Usually, student got confused between perpendicular and base. Always remember that out of three sides, perpendicular is that side which is opposite to the given angle. In following figure, perpendicular (P) is the side which is opposite to the purple shaded angle(given angle) Side opposite to the right angle (shaded red) is hypotenuse (H). Remaining side is known as base (B). Now, you must be knowing that all the trigonometric ratios can be derived from these sides. Let us see how. sin θ = P/H cos θ = B/H tan θ = P/B cot θ = B/P sec θ = H/B cosec θ = H/P Learning all these formulae is somewhat difficult because trigonometry is full of formulas. Let me tell you the very easy method to learn the above formulas. Following is the basic pattern which you should learn by heart. The easy way to remember this is the way our teachers teach us in schools. I am sure that is the best way to remember this. You can never forget this trick: (Some People Have) (Curly Brown Hair) (Through Proper Brushing) Learn the above line and write 3-3 words vertically like: Some Curly Through People Brown Proper Have Hair Brushing i.e. S C T P B P H H B sin cos tan P/H B/H P/B cot sec cosec B/P H/B H/P Now, It will seem very easy to learn the ratios. Isn't it? Basic Formula Following are some very basic formulae. Go through these formulas once, then we will proceed to the trigonometric ratios of standard angle. sec θ = 1/cos θ cosec θ = 1/sin θ cot θ = 1/tan θ tan θ = (sin θ)/cos θ sin²θ+ cos²θ = 1 1+tan²θ = sec²θ 1+ cot²θ = cosec²θ Trigonometric Ratios of Standard Angles You must have seen the following table of trigonometric ratios, but so many times you may get confused by remembering the values. Here, I will tell you the basic technique which will help you to learn this table. θ 0° 30° 45° 60° 90° sin θ 0 1/2 √2/2 √3/2 1 cos θ 1 √3/2 √2/2 1/2 0 tan θ 0 1/√3 1 √3 Not Defined cot θ Not Defined √3 1 1/√3 0 cosec θ Not Defined 2 √2 2/√3 1 sec θ 1 2/√3 √2 2 Not Defined Write numbers from 0 to 4 Firstly, to find the values of Row1 Divide the numbers with 4 Take the square root of all 0 → 0/4 → √0 = 0 1 → 1/4 → √1/4 = 1/2 2 → 2/4 → √2/4 = √2/2 3 → 3/4 → √3/4 = √3/2 4 → 4/4 = 1 See. how simple. You just need to divide 4 to numbers 0 to 4, then take square root. Row2 = It is just the reverse of Sin values. (See table) Row3 = Value of tan theta is derived by using ((sinθ)/(cosθ)) Row4 = Use cot(θ) = 1/tan(θ) to find cot values.