Question 4 of 18beginner🔧 ApplyNumerical2 marks

Given sec θ=1312\theta = \frac{13}{12}, calculate all other trigonometric ratios.

Correct Answer

Given sec θ = 13/12

Since sec θ = Hypotenuse/Base, let us consider a right triangle where Hypotenuse = 13k and Base = 12k.

Step 1: Find the perpendicular side using Pythagoras theorem Perpendicular² = Hypotenuse² - Base² = (13k)² - (12k)² = 169k² - 144k² = 25k² Therefore, Perpendicular = 5k

Step 2: Calculate all trigonometric ratios

  • cos θ = Base/Hypotenuse = 12/13
  • sin θ = Perpendicular/Hypotenuse = 5/13
  • tan θ = Perpendicular/Base = 5/12
  • cot θ = Base/Perpendicular = 12/5
  • cosec θ = Hypotenuse/Perpendicular = 13/5
Exercise: EXERCISE 8.1 | Q: 5 | (Page Page 9)
For More Understanding

Explanation

The textbook demonstrates that when one trigonometric ratio is known, we can construct a right triangle with sides in the given ratio and use Pythagoras theorem to find the third side. Since sec θ = Hypotenuse/Base = 13/12, we assign values 13k and 12k respectively. The perpendicular side is found using the Pythagoras theorem: 132−122\sqrt{13² - 12²} = 5. All other ratios are then calculated using their definitions from the triangle sides.

Solution Steps

  1. Step 1: Identify that sec θ = Hypotenuse/Base = 13/12, so let Hypotenuse = 13k and Base = 12k

  2. Step 2: Apply Pythagoras theorem: Perpendicular = Hypotenuse2−Base2\sqrt{Hypotenuse² - Base²} = 169k2−144k2\sqrt{169k² - 144k²} = 5k

  3. Step 3: Calculate cos θ = Base/Hypotenuse = 12/13

  4. Step 4: Calculate sin θ = Perpendicular/Hypotenuse = 5/13

  5. Step 5: Calculate tan θ = Perpendicular/Base = 5/12

  6. Step 6: Calculate cot θ = 1/tan θ = 12/5

  7. Step 7: Calculate cosec θ = 1/sin θ = 13/5

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