Trigonometry in Math

Trigonometry is a branch of mathematics that explores the relationships between the ratios of the sides of a right-angled triangle and its angles. The fundamental ratios used to study these relationships are known as trigonometric ratios, which include sine, cosine, tangent, cotangent, secant, and cosecant.

We use trigonometry in many everyday situations, often without even noticing. Here are some easy-to-understand examples:

• Construction and Architecture: Trigonometry helps calculate angles and heights when designing buildings, bridges, and roads. For example, architects use it to determine roof slopes or the angle of staircases.
• Navigation: Pilots and sailors use trigonometry to find their way. By calculating distances and angles between landmarks or stars, they can determine their location and the best route to follow.
• Engineering: Engineers use trigonometry in various fields, like designing machines, cars, or tunnels. For example, trigonometry ensures roads curve at safe angles for vehicles.

Trigonometry Concepts

• Trigonometric Ratios
• Trigonometry Table
• Trigonometry Formulas
• Trigonometric Functions
• Domain and Range of Trigonometric Functions
• Graph of Trigonometric Functions
• Application of Trigonometry in Real Life
• Height and Distance
• Trigonometric Equations
• Trigonometric Symbols

Trigonometry for Aptitude

• Trigonometry – Quiz
• Height and Distances – Aptitude Questions and Answers
• Height and Distances – Quiz for Aptitude
• Trigonometric Equations and Identities – Quiz
• Trigonometry for Non-Right Angles – Quiz

Trigonometry Practice Questions

• Trigonometry Practice Questions Easy
• Trigonometry Practice Questions Medium
  
• Trigonometry Practice Questions Hard
• Trigonometric Equations Practice Questions
• Trigonometric Identities Practice Problems
• Trigonometric Ratios Practice Questions

Trigonometry for Programming

• C++ Program to Illustrate Trigonometric Functions
• Trigonometric Functions in Java
• Trigonometric and Angular Functions in Python
• Trigonometric Functions in MATLAB
• Trigonometric Functions in LaTex

Trigonometry in Math – FAQs

What are the three types of trigonometry?

The three types of trigonometry are as follows:

• Core Trigonometry 
• Plane Trigonometry 
• Spherical Trigonometry 

What are the 6 ratios of trigonometry?

The six trigonometric ratios are sin, cos, tan, cot, sec, and cosec.

What are trigonometry identities?

An equation that holds true for all angles involving different trigonometric ratios is known as trigonometric identity. 

What is the value of sin 45?

The value of [Tex]\sin 45\degree \text is \frac{1}{\sqrt{2}} [/Tex]

Who invented trigonometry?

Trigonometry as a concept, was first introduced by an ancient Greek mathematician Hipparchus.

Where is trigonometry used in real life?

Trigonometry and its functions find diverse applications in our everyday lives. It’s instrumental in geography for measuring distances between landmarks, in astronomy for gauging distances to nearby stars, and crucial in satellite navigation systems for accurate positioning and mapping.

Suggested Quiz

10 Questions

If x × tan45 × cos60 = sin60 × cot60 , then x is:

• 1

• 0

• -1

• 2

Explanation:

x × 1 × 1/2 = √3/2 × 1/√3

⇒ x/2 = 1/2

⇒ x = 1

The value of cos 0°. cos 1°. cos 2°. cos 3°. . . cos 89° cos 90° is

• 1

• 2

• 0

• -1

Explanation:

Since the value of cos 90 is 0, so multiplication value is 0.

If sin(x) = 1/3, find cos(x/2)

• 2√2/3​

• √(3 + 2√2)/6

• √(2 + 3√2)/6

• None of These

Explanation:

Given: sin(x) = 1/3

cos(x) = 2√2/3

Use the half-angle formula for cosine: cos(θ/2) = (√(1+cos(θ))/2)

cos(x/2) = √(1+cos x)/2 = √(1 + 2√2/3)/2 = √(3 + 2√2)/6

What is the maximum value of 3 Sinθ + 4 cosθ?

• 12

• 5

• 6

• 1

Explanation:

(minimum value)-√(a² + b²) ≤ a Sinθ + b cosθ ≤ √(a² + b²) (maximum value)

Thus, Maximum value of 3 Sinθ + 4 cosθ = √(3² + 4²) = 5

What is minimum value of Sinθ + cosθ ?

• -2

• √3/2

• -1

• -√2

Explanation:

(minimum value)-√(a² + b²) ≤ a Sinθ + b cosθ ≤ √(a² + b²) (maximum value)

Thus, minimum value of sin θ + cos θ = -√(1² + 1²) = -√2

If tan (x+y) tan (x-y) = 1, then find tan (2x/3)?

• 1/√3

• 1/2

• 1/√2

• 2/√3

Explanation:

tan A × tan B = 1
then, tan A = cot B,
So, A + B = 90^o

(x + y) + (x - y) = 90^o,
⇒ 2x = 90^o,
⇒ x = 45^o

Thus, tan (2x/3) = tan 30^o = 1/√3

A man is watching from the top of a tower a boat speeding away from the tower. The boat makes an angle of depression of 45° with the man's eye when at a distance of 60 meters from the tower. After 5 seconds, the angle of depression becomes 30°. What is the approximate speed of the boat, assuming that it is running in still water?

• 32 kmph

• 36 kmph

• 40 kmph

• 44 kmph

Explanation:

Let the tower be PQ and the boat be at positions R and S when making angles of 45° and 30° respectively. 
Given, PR = 60 m.
Now, PQ/PR = tan 45° = 1.
So, PQ = PR = 60 m. 
Again, PQ/PS = tan 30° = 1/√3.
So, PS = 60 × √3 m = 103.92 m. 
Distance covered in 5 seconds = 103.92 - 60 = 43.92 m.
Speed in kmph = (43.92/5) × (18/5) = 32 kmph (approximately).

The value of sin²10 + sin²20 + sin²30 + . . . + sin²80 is equal to

• 0

• 8

• 3

• 4

Explanation:

sin²10 + sin²20 + sin²30 + . . . + sin²80

= sin²10 + sin²80 + sin²20 + sin²70 + sin²40 + sin²50 + sin²30 + sin²60

= (sin²10 + cos²10) + (sin²20 + cos²20) + (sin²40 + cos²40) + (sin²30 + cos²30) [As cos (90 - x) = sin x]

= 1 + 1 + 1 + 1 = 4 [As sin²x + cos²x = 1]

If cos⁴A - sin⁴A = p, then the value of p is:

• 2 cos²A - 1

• 2 cos²A + 1

• 2 cos²A - 2

• 2 cos²A + 2

Explanation:

cos⁴A - sin⁴A = p

⇒ (cos²A + sin²A)(cos²A - sin²A) = p

⇒ (cos²A - sin²A) = p [As we know, cos²A + sin²A = 1]

⇒ cos²A - (1 - cos²A) = p [As sin²A = 1 - cos²A]

⇒ cos²A - 1 + cos²A = p

⇒ 2cos²A - 1 = p

Amit is standing at a point P is watching the top of a tower, which makes an angle of elevation of 45° with Amit's eye. He walks some distance towards the tower to watch its top and the angle of elevation becomes 60°.

What is the distance between the base of the tower and the point P?

• 4.2 units

• 8 units

• 10 units

• Data inadequate

Explanation:

Let MN be the tower and Amit be standing at P (45° = angle MPN)
and Q (60° = angle MQN).

We are only given two angles and no sides of the triangles, thus we can't find any sides. Therefore, the data is inadequate.

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