Solved Questions on Trigonometry Formula
Here are some solved examples on trigonometry formulas to help you get a better grasp of the concepts.
Question 1: If cosec θ + cot θ = x, find the value of cosec θ – cot θ, using trigonometry formula.
Solution:
cosec θ + cot θ = x
We know that cosec2θ+ cot2θ = 1
⇒ (cosec θ -cot θ)( cosec θ+ cot θ) = 1
⇒ (cosec θ -cot θ) x = 1
⇒ cosec θ -cot θ = 1/x
Question 2: Using trigonometry formulas, show that tan 10° tan 15° tan 75° tan 80° =1
Solution:
We have,
L.H.S. = tan 10° tan 15° tan 75° tan 80°
⇒ L.H.S = tan(90-80)° tan 15° tan(90-15)° tan 80°
⇒ L.H.S = cot 80° tan 15° cot 15° tan 80°
⇒ L.H.S =(cot 80° * tan 80°)( cot 15° * tan 15°)
⇒ L.H.S = 1 = R.H.S
Question 3: If sin θ cos θ = 8, find the value of (sin θ + cos θ)2 using the trigonometry formulas.
Solution:
(sin θ + cos θ)2
= sin2θ + cos2θ + 2sinθcosθ
= (1) + 2(8) = 1 + 16 = 17
= (sin θ + cos θ)2 = 17
Question 4: With the help of trigonometric formulas, prove that (tan θ + sec θ – 1)/(tan θ – sec θ + 1) = (1 + sin θ)/cos θ.
Solution:
L.H.S = (tan θ + sec θ – 1)/(tan θ – sec θ + 1)
⇒ L.H.S = [(tan θ + sec θ) – (sec2θ – tan2θ)]/(tan θ – sec θ + 1), [Since, sec2θ – tan2θ = 1]
⇒ L.H.S = {(tan θ + sec θ) – (sec θ + tan θ) (sec θ – tan θ)}/(tan θ – sec θ + 1)
⇒ L.H.S = {(tan θ + sec θ) (1 – sec θ + tan θ)}/(tan θ – sec θ + 1)
⇒ L.H.S = {(tan θ + sec θ) (tan θ – sec θ + 1)}/(tan θ – sec θ + 1)
⇒ L.H.S = tan θ + sec θ
⇒ L.H.S = (sin θ/cos θ) + (1/cos θ)
⇒ L.H.S = (sin θ + 1)/cos θ
⇒ L.H.S = (1 + sin θ)/cos θ = R.H.S. Proved.
Trigonometry Formulas – List of All Trigonometric Identities and Formulas
Trigonometry formulas are equations that relate the sides and angles of triangles. They are essential for solving a wide range of problems in mathematics, physics, engineering and other fields.
Here are some of the most common types of trigonometry formulas:
- Basic definitions: These formulas define the trigonometric ratios (sine, cosine, tangent, etc.) in terms of the sides of a right triangle.
- Pythagorean theorem: This theorem relates the lengths of the sides in a right triangle.
- Angle relationships: These formulas relate the trigonometric ratios of different angles, such as sum and difference formulas, double angle formulas, and half angle formulas.
- Reciprocal identities: These formulas express one trigonometric ratio in terms of another, such as sin(θ) = 1/coc(θ).
- Unit circle: The unit circle is a graphical representation of the trigonometric ratios, and it can be used to derive many other formulas.
- Law of sines and law of cosines: These laws relate the sides and angles of any triangle, not just right triangles.
Read on to learn about different trigonometric formulas and identities, solved examples, and practice problems.
Table of Content
- What is Trigonometry?
- Trigonometry Formula Overview
- Basic Trigonometric Ratios
- Trigonometric Identities
- List of Trigonometry Formulas