What is sec θ , cosec θ, tan θ?
In mathematics, sec, cosec, and tan are abbreviations for trigonometric functions. They are related to angles and sides of right triangles and are fundamental in trigonometry.
1. Secant (sec): The secant of an angle in a right triangle is the ratio of the length of the hypotenuse to the length of the adjacent side. It is the reciprocal of the cosine function.
sec (θ)= 1/cos (θ)
Example: Let's say you have a right triangle with an angle of 30 degrees. If the length of the adjacent side (the side adjacent to the angle) is 4 units, and the length of the hypotenuse is 5 units, then:
sec (30)= 5/4
2. Cosecant (cosec): The cosecant of an angle in a right triangle is the ratio of the length of the hypotenuse to the length of the opposite side. It is the reciprocal of the sine function.
csc (θ)= 1/sin (θ)
Example If in the same triangle mentioned above, the length of the opposite side (the side opposite to the angle) is 3 units, then:
csc 30 = 5/3
3. Tangent (tan): The tangent of an angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side.
tan θ = opposite side/ adjacent side
Example: Considering the same triangle, the tangent of the angle 30 degrees would be:
tan (30) = 3/4
These trigonometric functions find extensive applications in various fields such as engineering, physics, astronomy, and more. They are particularly useful in problems involving angles and distances in triangles or circular motion.
Question :- Prove cos(θ) + tan(θ) = 1
To prove that cos(θ) + tan(θ) = 1, we'll use trigonometric identities. One way to approach this is by expressing tangent in terms of sine and cosine, and then combining like terms.
First, recall the definition of tangent:
tan(θ) = sin(θ)/cos(θ)
Now, let's substitute this expression for tangent into the equation:
cos(θ) + tan(θ) = cos(θ) + sin(θ)/cos(θ)
To combine these terms, we'll make the denominator of the second term the same as the denominator of the first term by multiplying the numerator and the denominator of the second term by (cos(θ)):
= cos(θ) + sin(θ). cos(θ) /cos(θ). cos(θ)
= cos(θ) + /sin(θ). cos(θ)cos^2(θ)
Now, recall the Pythagorean identity for sine and cosine:
sin^2(θ) + cos^2(θ) = 1
We can rewrite cos^2(θ) as 1 - sin^2(θ):
= cos(θ) + sin(θ) . 1 - sin^2(θ) / 1 - sin^2(θ)
= cos(θ) + sin(θ) - sin^3(θ)/1 - sin^2(θ)
Now, using the Pythagorean identity again, we know that 1 - sin^2(θ) = cos^2(θ):
= cos(θ) + sin(θ) - sin^3(θ)/cos^2(θ)
= cos(θ) + sin(θ)/cos^2(θ) - sin^3(θ)/cos^2(θ)
= cos(θ) + sin(θ)/cos(θ) - tan^2(θ)
Now, using the fact that tan(θ) = sin(θ)/cos(θ), we can replace the first two terms with tan(θ):
= tan(θ) - tan^2(θ)
Now, let's apply another trigonometric identity:
tan^2(θ) + 1 = sec^2(θ)
So,
tan^2(θ) = sec^2(θ) - 1
Substitute this into our equation:
= tan(θ) - (sec^2(θ) - 1)
= tan(θ) - sec^2(θ) + 1
Now, remember that sec(θ) = /1cos(θ):
= tan(θ) – 1/cos^2(θ) + 1
= tan(θ) – {cos^2(θ)/cos^2(θ)} + 1
= tan(θ) - tan^2(θ) + 1
= 1
Therefore, cos(θ) + tan(θ) = 1.
