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High School Mathematics - 2
10.3 Trignometric Ratios - Sin θ, Cos θ and Tan θ

Introduction:
  • Right triangles are of importance in geometry. Thus, mathematicians have developed a shorthand for writing the ratios of the sides of right triangles.
  • Instead of writing "the ratio of the leg adjacent to an 18 degree angle to the hypotenuse of the triangle" we write cos 18.
  • Because expression of this type frequently come up in physics, engineering and many other branches of science, such a shorthand was developed.
Sine, Cosine, Tangent of an Angle


In the right angled triangle ΔPAB, ∠PAB = θ
Then the ratio

PB/AP is called the sine of angle θ.

It is written as



  • In other form



  • Note : If ∠APB = φ then


    AB/AP is called the cosine of angle θ

    It is written as




  • In other form



  • Note : If ∠APB = φ then

    PB/AB is called the tangent of angle θ.
    It is written as



  • In the other form


  • or


An easy way to remember these relationships for trig functions and the right triangle is given below
Write down this mnemonic:
SOH - CAH - TOA
It is pronounced "so - ka - toe - ah".
The SOH stands for "Sine of an angle is Opposite over Hypotenuse."
The CAH stands for "Cosine of an angle is Adjacent over Hypotenuse."
The TOA stands for "Tangent of an angle is Opposite over Adjacent".

Directions: Answer the following questions. Also write at least 10 examples of your own.
Q 1: ΔABC has right angle at A. Write down sin B, cos C and tan B if BC = 2, AB = AC = 1
sin B = 1/2; cos C = 1; tan B = 1
sin B = 1; cos C = 1; tan B = 2
sin B = 1; cos C = 1/2; tan B = 1
sin B = 1/2; cos C = 1/2; tan B = 1

Q 2: ΔABC has right angle at B. Write down sin A and tan A if BC = 5 cm and AB = 12 cm
sin A = 12/13; tan A = 12/5
sin A = 5/13; tan A = 5/12
sin A = 5/12; tan A = 5/13
sin A = 12/13; tan A = 5/12

Q 3: ΔABC has right angle at A. Write down sin B, cos C and tan B if AB = 20 cm, AC = 21 cm
sin B = 21/29; cos C = 20/29; tan B = 21/20
sin B = 20/29; cos C = 20/29; tan B = 20/29
sin B = 21/29; cos C = 21/29; tan B = 21/20
sin B = 20/29; cos C = 21/29; tan B = 21/20

Q 4: From the following figure, find sin θ, cos θ, tan φ.

sin θ = 3/5, cos θ = 4/5, tan φ = 3/4.
sin θ = 3/5, cos θ = 4/5, tan φ = 4/3.
sin θ = 4/5, cos θ = 3/5, tan φ = 4/3.
sin θ = 3/5, cos θ = 1, tan φ = 4/3.

Q 5: In the figure, ΔABC is right angled at C. Find cos A and tan B

cos A = 7/25, tan B = 7/24
cos A = 1, tan B = 7/24
cos A = 24/25, tan B = 7/24
cos A = 7/25, tan B = 1

Q 6: In a triangle ABC, right angled at B, if BC = 12 and AB = 9 then find cos C.
cos C = 9/12
cos C = 12/15
cos C = 12/25
cos C = 1

Q 7: In a triangle ABC, right angled at C, AC = m and BC = (m2-1)/2. Find sin A.
sin A = 2m/(m2+1)
sin A = m/(m2+1)
sin A = (m2-1)/(m2+1)
sin A = 2m/(m2-1)

Q 8: In a right angled triangle, the sides other than the hypotenuse are equal and measure 2 cm each. Find sin 45.
1
1/2
2
0

Question 9: This question is available to subscribers only!

Question 10: This question is available to subscribers only!


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