Trigonometry (Basics, Formulas, Identities, Question and Solution)
Trigonometry word is formed from the ancient Greek words “trigonon” and “metron” which mean triangle and measure respectively, thus collectively Trigonometry mean measures of a triangle. Historians believe that in ancient Greece, a mathematician named Hipparchus was the first to introduce the idea of trigonometry by giving the first tables of chords which is the modern-day equivalent to the table of values of trigonometric ratio sine. Other than Greece, roots of this subject are also found in ancient India where Aryabhatta (an Indian mathematician and astronomer) documented the modern intuition of trigonometric ratios.
What is Trigonometry?
Trigonometry is the branch of mathematics that deals with the side and angles of a triangle, especially the right angle triangle i.e., a triangle with one angle right angle. Other than right angles, trigonometry is helpful in many different geometric figures, either 2-dimensional or 3-dimensional. Studies of Trigonometry can be classified into three parts which are as follows:
- Core Trigonometry (deal with right angle triangles only)
- Plane Trigonometry (deals with all types of 2-dimensional geometry)
- Spherical Trigonometry (deals with all types of 3-dimension geometry)
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Trigonometry Basics are the core concepts of trigonometry without which it can’t be defined, some of these basics of trigonometry are as follows:
- Angles: The measure of space between two intersecting lines are known as angles.
- Right-angle Triangle: A triangle with one of its interior angles being the right angle i.e., 90°, is called right angles triangle.
- Pythagoras Theorem: In right angles triangle, according to the Pythagoras theorem, the square of the hypotenuse is equal to the sum of squares of the other two sides,
- Trigonometric Ratios: Trigonometric Ratios are defined as the ratio of the sides of the right angle triangles. As there are 3 ways to choose two sides out of three and two ways for each chosen pair to arrange in ratio, thus there are 3×2 =6 trigonometric ratios which are defined for each possible pair of sides of the right angle triangle.
Important Trigonometry Functions
The important trigonometric functions include sin and cos, as all the other trigonometric ratios can be defined in terms of sin and cos.
- tangent or tan can be defined as the ratio of sin and cos i.e., tan x = sin x/cos x,
- cotangent or cot can be defined as the ratio of cos and sin i.e., cot x = cos x/sin x, and
- cosecant or cosec is the inverse of the sin .i.e, cosec x = 1/sinx,
- secant or sec is the inverse of cos i.e., sec x = 1/cos x.
That’s why the only two important trigonometric ratios are sin and cos.
In the study of Trigonometry, there are six trigonometric ratios i.e., sin, cos, tan, cot, sec, and cosec. These trigonometric ratios as the name suggest defined as the ratios between two sides of a right-angle triangle. For the sin trigonometric ratio, we defined it as the ratio of the perpendicular (opposite to the angle for which we are calculating sin,) to the hypotenuse of the right-angle triangle i.e., for triangle ABC right angles at B,
Similarly, cos is defined as the ratio of the base and hypotenuse of the right-angle triangle,
and, trigonometric ratio tan is defined as the ratio of perpendicular to the base of the right-angle triangle.
Other, then sin, cos and tan, sec, cosec and cot are also defined as the ratio of the sides of right angle triangle as follows:
sec, cosec, and cot are the reciprocals of the trigonometric ratios cos, sin, and tan respectively.
Trigonometric Table is the table of values of trigonometric ratio at some standard angle values. The table for the value of trigonometric ratios at different angles is given below:
To create a chart of different values of trigonometric ratios at different angles, we can use the following algorithm. Using this trick we can complete the trigonometry chart without remembering the exact values of the ratios for different angles. The steps required in this algorithm are as follows:
Write first five whole numbers with some distance between them.
0 | 1 | 2 | 3 | 4
Divide each number by 4.
0 | 1/4 | 2/4 | 3/4 | 4/4
0 | 1/4 | 1/2 | 3/4 | 1
Take Square Root for each resulting number in step 2.
√0 | √(1/4) | √(1/2) | √(3/4) | √1
0 | 1/2 | 1/√2 | √(3)/2 | 1
Resulting values for step 3, are the value of trigonometric ratio sine for angles 0°, 30°, 45°, 60°, 90°.
Reverse the order of the resulting values in step 3, to get the value of cos for the same angles.
1 | √(3)/2 | 1/√2 | 1/2 | 0
Find the ratio of the results in step 3 to results in step 5, to get the value of cos for angles 0°, 30°, 45°, 60°, 90°.
0÷1 | 1/2÷√(3)/2 | 1/√2÷1/√2 | √(3)/2÷1/2 | 1÷0
0 | 1/√3 | 1 | √(3) | Not defined
Thus, using these steps, resulting table is formed.
The angle for which the trigonometric ratio is defined is the trigonometric angle. Angles can either be measured in degrees (°) or can be measured in radians (rad). Some standard angles for which we create the table of trigonometric ratios are 0°, 30°, 45°, 60°, and 90°. Other than these angles, we also sometimes need to deal with 15°, 18°, 75°, and 72°.
Application of Trigonometry
Trigonometry has so many applications in the real world, we can even say that is the most used mathematics concept throughout mathematics. Some of the applications of trigonometry are as follows:
- Trigonometry is very essential for modern-day navigation systems such as GPS or any other similar system.
- In most streams of engineering, trigonometry is used extensively for various kinds of analysis and calculations, which helps engineers to make more sound decisions for the construction of various kinds of structures.
- Various trigonometric concepts and formulas are used in the computer graphics of the modern age, as computer graphics are created in 3-D environments so all the calculations are done by the graphics processing unit of the computer to deliver the computer graphics as output.
- Various astronomical calculations such as the radius of celestial bodies, the distance between objects, etc. involve the use of trigonometry and its different trigonometric ratios.
- In Physics, we use trigonometry to understand and evaluate many real-world systems such as the orbits of planets and artificial satellites, the reflection or refraction of light ni various environments, etc.
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Trigonometry Formulas List
In trigonometry, there are a lot of formulas or identities that relate the different trigonometric ratios with each other for different values of angles. Some of these identities are:
Pythagorean Trigonometric Identities
- sin2 θ + cos2 θ = 1
- 1+tan2 θ = sec2 θ
- cosec2 θ = 1 + cot2 θ
Sum and Difference Identities
- sin (A+B) = sin A cos B + cos A sin B
- sin (A-B) = sin A cos B – cos A sin B
- cos (A+B) = cos A cos B – sin A sin B
- cos (A-B) = cos A cos B + sin A sin B
- tan (A+B) = (tan A + tan B)/(1 – tan A tan B)
- tan (A-B) = (tan A – tan B)/(1 + tan A tan B)
Double Angle Identities
- sin 2θ = 2 sinθ cosθ
- cos 2θ = cos2θ – sin 2θ = 2 cos 2 θ – 1 = 1 – sin 2 θ
- tan 2θ = (2tanθ)/(1 – tan2θ)
Half Angle Identities
Product Sum Identities
Triple Angle Identities
Euler’s Formula for Trigonometry
For the imaginary power of exponent e(Euler’s number), Euler gave an identity that relates the imaginary power of e to the trigonometric ratios sin and cos, the identity is given as follows:
is the imaginary number which is defined as i = √(-1)
, and Φ
is the angle.
Putting, -Φ instead of Φ in the above identity, we get
Now, adding and subtracting these two values together we get, values of sin and cos in terms of imaginary power of Euler’s number,
Practice Questions : For Trigonometry
There are many real-life examples where trigonometry is used broadly. If we have been given with height of the building and the angle formed when an object is seen from the top of the building, then the distance between object and bottom of the building can be determined by using the tangent function, such as tan of angle is equal to the ratio of the height of the building and the distance. Let us say the angle is ∝, then:
Tan ∝ = Height/Distance between object & building
Distance = Height / Tan ∝
Let us assume that height is 20m and the angle formed is 45 degrees, then
Distance = 20 / Tan 45°
Since, tan 45° = 1
So, Distance = 20 m
Two friends, Rakesh and Vishal started climbing a pyramid-shaped hill. Rakesh climbs 315 m and finds that the angle of depression is 72.3 degrees from his starting point. How high is he from the ground?
Let m is the height above the ground.
To find: Value of m
To solve m, use the sine ratio.
Sin 72.3° = m/315
0.953 = m/315
m= 315 x 0.953
The man is 300.195 mtr above the ground.
Question 3: A man is observing a pole of height 55 foot. According to his measurement, pole cast a 23 feet long shadow. Can you help him to know the angle of elevation of the sun from the tip of shadow?
Solution: Let x be the angle of elevation of the sun, then
tan x = 55/23 = 2.391
x = tan-1(2.391)
or x = 67.30 degrees
Question 4: A ladder is leaning against a wall. The angle between the ladder and the ground is 45 degrees, and the length of the ladder is 10 meters. How far is the ladder from the wall?
Solution: Let the distance between the ladder and the wall be x meters.
Here, ladder, wall and ground together makes a right angle triangle, where for given angle,
Length of ladder = hypotenous = 10 meter,
Distance between wall and laddar = base = x meter
Using trigonometric ratio cos, we get
⇒ cos(45°) = = x/10
⇒ cos(45°) = 1
⇒1/√2 = x/10
⇒ x = 10/√2 = 5√2 meters
Therefore, the ladder is 5√2 meters away from the wall.
Question 5: A right-angled triangle has a hypotenuse of length 10 cm and one of its acute angles measures 30°. What are the lengths of the other two sides?
Solution: Let’s call the side opposite to the 30° angle as ‘a’ and the side adjacent to it as ‘b’.
Now, sin (30°) = perpendicular/hypotenous = a/10
⇒ a = 10 × sin(30°) = 5 cm [sin(30°) = 1/2]
and cos(30°) = b/10
⇒ b = 10 × cos(30°) = 10 × √(3)/2 ≈ 8.66 cm
Therefore, the lengths of the other two sides are 5 cm and 8.66 cm (approx.).
Question 6: Prove that (cos x/sin x) + (sin x/cos x) = sec x × cosec x.
Solution: LHS = (cos x/sin x) + (sin x/cos x)
⇒ LHS = [cos2x + sin2x]/(cos x sin x)
⇒ LHS = 1/(cosx sinx) [Using cos2x + sin2x = 1]
⇒ LHS = (1/cosx) × (1/sinx)
⇒ LHS = secx × cosecx = RHS [ 1/cosx = sec x and 1/sinx = cosec x]
Question 7: A person is standing at a distance of 10 meters from the base of a building. The person measures the angle of elevation to the top of the building as 60°. What is the height of the building?
Solution: Let h be the height of the building.
And, all the distances here in the question make a right angle triangle, with a base of 10 meters and height h meter.
As tan θ = Perpendicular/Base
⇒ tan(60°) = h / 10
⇒ h = 10 tan(60°)
Using the values of tan(60°) = √3, we get:
h = 10√3 ≈ 17.32 m
Therefore, the height of the building is approximately 17.32 meters.
Question 8: Find the value of x in the equation cos-1(x) + sin-1(x) = π/4.
Solution: For, cos-1(x) + sin-1(x) = π/4
As we know, cos-1(x) + sin-1(x) = π/2, above equation becomes
π/2 = π/4, which is not true.
Thus, the given equation has no such value of x, which can satisfy the equation.