Mathematics

Trigonometry - Sine, Cosine and Tangent explained with worked examples

len Alfred Ajibola - 12th February, 2019 @ 03:05

Topics in Mathematics

Angles: Type of angles based on degree measurement Angle Pairs: Types of angle pairs Conversion of Number Bases explained with examples Angles: What are Angles? Categories of Angles Lines and Types in Mathematics Number Base System - Table showing the Number Bases with explanations Mathematics Scheme of Work, SS1, 3rd Term Mathematics Scheme of Work, SS1, 2nd Term Mathematics Scheme of Work, SS1, 1st Term Trigonometry - Sine, Cosine and Tangent explained with worked examples Quadratic Equation - Factorization of Quadratic Equation Algebra - Explanation and Worked Examples on Algebra


Academic Questions in Mathematics

Please check out our Test Your Knowledge page to see all Questions and Answers

Factorize 12a2 - 20a + 3

  • A. (6a - 1)(2a - 3)

  • B. (6a - 1)(2a + 3)

  • C. (6a + 1)(2a - 3)

  • D. (6a + 1)(2a + 3)

  • E. (-6a + 1)(2a - 3)

  • F. (-6a + 1)(2a + 3)

Simplify 10ab + 7ab

  • A. 17ab2

  • B. 17a2b2

  • C. 3ab2

  • D. 3a2b2

  • E. 17ab

  • F 3ab

Which of the following is not a type of angle?

  • A. Right angle

  • B. Left Angle

  • C. Acute Angle

  • D. Complete Angle

  • E. Straight Angle

  • F. Reflex angle

Two lines that can never meet each other are best considered as _____ lines.

  • A. straight
  • B. vertical
  • C. horizontal
  • D. parallel
  • E. oblique
  • F. perpendicular

An angle at exactly 90o is a/an _____ angle.

  • A. acute
  • B. obtuse
  • C. reflex
  • D. right
  • E. complete
  • F. straight

Slanted or tilted lines that will finally meet at some point but NOT at a right angle are referred to as _____.

  • A. Oblique
  • B. Curved
  • C. Vertical
  • D. Perpendicular
  • E. Parallel
  • F. Horizontal

The two sides that forms an angle are referred to as the _____.

  • A. Arms
  • B. Vertexes
  • C. Bays
  • D. Zones
  • E. Limbs
  • F. Dune

The point where the two sides of an angle meet is termed the _____.

  • A. Arms
  • B. Vertex
  • C. Bays
  • D. Zones
  • E. Limbs
  • F. Dune

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Sine Cosine and Tangent:

Trigonometry is an aspect of mathematics that deals with triangles and the relationship between the 'three sides' and 'three angles' of triangles.

In Trigonometry, the right triangle is of interest to us because sine 'sin (θ)' and cosine 'cos (θ)' and tangent 'tan (θ)' are the three functions that reveal their shapes.

Please read on Angles and Types of Angles here.

Before exploring these functions, let’s understand the names given to the three sides of a right triangle:

Sides of a right angle triangle - Len Academy


Note: The angle of the above right triangle is denoted as θ

Hypotenuse is always the longest side of the triangle.

Opposite is the side opposite the angle (θ)

Adjacent is the shorter side next to the angle (θ)

Each of Sin θ, cos θ and tan θ are the ratio of sides of the right triangle. They are calculated by using the formula below:

Sohcahtoa - Len Academy

 

To get the ratio of the angle, make sure to always remember the magic word SOHCAHTOA

  • SOH
  • Sin = Opposite/Hypotenuse


  • CAH

  • Cos = Adjacent/Hypotenuse


  • TOA

  • Toa = Opposite/Adjacent


Note: The functions sine, cosine and tangent helps us to easily calculate the sides of the triangle (, Opposite and Adjacent) when we know the angle.

Similarly, it also aid us to calculate the angle (sine, cosine and tangent) when we know the sides of the triangle.

Please read on Lines in Mathematics here.

Below are some worked Examples:

 

Example 1

Find the sine, cosine and tangent of 30° whose hypotenuse has a value of 2, opposite has a value of 1 and adjacent has a value of √3.

The image below illustrates the question.

Len Academy

Since we already know the length of each side of the triangle, the angles for sine, cosine and tangent can easily be calculated by implementing the keyword SOH-CAH-TOA

Sine = Opposite/Hypotenuse

Sin(30°) = 1 / 2 = 0.5

Answer: Sin(30°) is 0.5

 

Cosine = Adjacent/Hypotnuse

Cos(30°) = 1.732 / 2 = 0.866

Answer: Cos(30°) is 0.866

 

Tangent

Tan(30°) = 1 / 1.732 = 0.577

Answer: Tan(30°) is 0.577

Note: You may confirm the answers from a scientific calculator if you wish.

Please read on Algebra explained with some worked examples.

 

Example 2

Find “d” from the diagram below using the sine function.

Len Academy
Image Credit: Maths is Fun

From the figure above, we can conclude that:

  • The angle is 39o

  • The hypotenuse is 30m (Longest side of the triangle)

  • d” is the opposite side of the triangle (side opposite the angle)

Applying the SOH-CAH-TOA formula, we will get:

Sin 39° = Opposite/Hypotenuse

Sin 39° = d/30

d = Sin 39° x 30

d = 0.6293 x 30

Answer: d = 18.9cm

Note: Once we know the value of sine, cosine and tangent in a right triangle, we will be able to calculate the less common functions (Secant, Cosecant and Cotangent) using the formula below:

Secant Function:

  sec(θ) = 1/cos

Cosecant Function:

  csc(θ) = 1/sin

Cotangent Function:

  cot(θ) = 1/tan

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Alfred Ajibola is a Medical Biochemist, a passionate Academician with over 7 years of experience, a Versatile Writer, a Web Developer, a Cisco Certified Network Associate and a Cisco CyberOps Associate.


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Amazing facts in Mathematics

The opposite sides of a 🎲 dice 🎲 will always give a sum of seven

31,556,926 seconds makes one year

From 0 through 1000, the only number that has the letter 'a' in it is 'one thousand'

÷

The division symbol '÷' is actually called an obelus

Four is the only number in the English language that is spelt with the same number of letters as the number (4) itself

40 is spelt as FORTY and not FOURTY. It is the only number that is spelt with letters arranged in alphabetical order 🙄

An odd number is a number that cannot be divisible by 2.

Every odd number has an 'e' in its spelling. They include:

  • One
  • Three
  • Five
  • Seven
  • Nine

555 in Thailand should sound like 'hahaha'. This is because the word '5' means 'ha' in Thailand

10! Actually means 10 factorial in mathematics. Below is its calculation:

10! = 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 3628800

3628800 seconds is the same as 42 days

42 days is the same as 6 weeks

We can therefore say that 10! equals 6 weeks or 3628800 seconds in mathematics

When you multiply 111,111,111 by 111,111,111; the result equals 12,345,678,987,654,321


NOTABLE POINTS IN Mathematics

Sides of a right angle triangle - Len Academy

 

The image above shows a right angled triangle. The angle of the above right triangle is denoted as θ

 

Hypotenuse is always the longest side of the triangle.

Opposite is the side opposite the angle (θ)

Adjacent is the shorter side next to the angle (θ)

Each of Sin θ, cos θ and tan θ are the ratio of sides of the right triangle. They are calculated by using the formula (margic word) sohcahtoa:

Sohcahtoa - Len Academy

soh

Sin θ = Opposite/Hypotenuse

cah

Cos θ = Adjacent/Hypotenuse

toa

Tan θ = Opposite/Adjacent

Please read more on sine, cosine and tangent here

Circumference in mathematics means the distance around the edge of a circle (or a another curverd surface).

Perimeter means the distance around a square or rectangle or polygon.

Circumference is to circle or any other curved surface

Circumference of a Circle - Len Academy

Perimeter is to square or rectangle or polygon

Perimeter of a Rectangle - Len Academy