**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.

**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)

**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.

- 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.

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:

Thus, using these steps, resulting table is formed.

- 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.

- sin
^{2}θ + cos^{2}θ = 1 - 1+tan
^{2}θ = sec2 θ - cosec
^{2}θ = 1 + cot^{2}θ

- 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)

- sin 2θ = 2 sinθ cosθ
- cos 2θ = cos
^{2}θ – sin^{2}θ = 2 cos^{2 }θ – 1 = 1 – sin^{2}θ - tan 2θ = (2tanθ)/(1 – tan
^{2}θ)

where,

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,

and

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

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

m=300.195 mtr

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 = [cos^{2}x + sin^{2}x]/(cos x sin x)

⇒ LHS = 1/(cosx sinx) [Using cos^{2}x + sin^{2}x = 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.**

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