In mathematics, trigonometry is one of the most important topics to learn. Trigonometry is the study of triangles, where "Trigon" means triangle and "metry" means measurement. Below is a list of formulas based on the right triangle that can be used for studying trigonometry.
The three main functions in trigonometry are Sine, Cosine and Tangent. They are just the length of one side divided by another
Consider a rightangled triangle with an angle θ, a hypotenuse, a side adjacent to angle θ, and a side opposite the angle to angle θ. For a given angle θ each ratio stays the same no matter how big or small the triangle is
When we divide Sine by Cosine we get:
Secant Function: sec(θ) = Hypotenuse / Adjacent
Cotangent Function: cot(θ) = Adjacent / Opposite
The trigonometric table is simply a collection of the values of trigonometric ratios for various angles including 0°, 30°, 45°, 60°, 90°, sometimes with other angles like 180°, 270°, and 360° included, in a tabular format.
Here is the trigonometry table for standard angles along with some nonstandard angles:
Tangent and Cotangent Identities
Reciprocal Identities
Pythagorean Identities
Even and Odd Angle Formulas
Cofunction Formulas
Important Trigonometric Ratios
If θ is any angle , then  θ, 90 ±θ , 180 ± θ, 270 ± θ , 360 ± θ etc. are called ALLIED ANGLES.
Example 1. Express 1·2 radians in degree measure.
1·2 radians =
Example 2. Calculate sin α if cos α =  9/11 and α ∈ (π, 3π/2).
For any angle a belonging to the indicated interval sin a is negative, and therefore,
Example 3. Calculate tan α if cos α = √5/5 and α ∈ (π, 3π/2)
For any angle a belonging to the indicated interval tanα is positive and cos a is negative, and therefore
Factorisation of the Sum Or Difference of Two sines Or cosines:
Transformation of Products Into Sum Or Difference Of sines & cosines:
Example 4. Suppose x and y are real numbers such that tan x + tan y = 42 and cot x + cot y = 49. Find the value of tan(x + y).
tan x + tan y = 42 and cot x + cot y = 49
now, cot x + cot y = 49
Example 5. If x sinθ = y sin(θ + 2π/3) = z sin (θ + 4π/3) then:
(a) x + y + z = 0
(b) xy + yz + zx = 0
(c) xyz + x + y + z = 1
(d) none
Correct Answer is Option (b)
= xy + yz + zx = 0
Example 6. Find θ satisfying the equation, tan 15° . tan 25° . tan 35° = tan 0, where θ ∈ (0, 15°).
LHS = tan 15° . tan (30°  5°) . tan (30° + 5°)
let t = tan 30° and m = tan 5°
Example 7. If tan A & tan B are the roots of the quadratic equation, a^{ }x^{2} + b^{ }x + c = 0 then evaluate a sin^{2} (A + B) + b sin (A + B) . cos (A + B) + c cos^{2} (A + B).
Now E = cos^{2} (A + B) [a tan^{2} (A + B) + b tan (A + B) + c]
Example 8._{ }Show that cos^{2}A + cos^{2}(A + B) + 2 cosA cos(180° + B) · cos(360° + A + B) is independent of A. Hence find its value when B = 810°.
cos^{2}A + cos^{2}(A + B)  [2 cosA · cosB · cos (A + B)]
= cos^{2}A + cos^{2}(A + B)  [ {cos(A + B) + cos(A  B) } cos (A + B) ]
= cos^{2}A + cos^{2}(A + B)  cos^{2}(A + B)  (cos^{2}A  sin^{2}B)
= sin^{2}B which is independent of A
now, sin^{2}(810°) = sin^{2}(720° + 90°) = sin^{2}90° = 1
If A+B+C = π
Example 10. If A + B + C = π, prove that,
Solution:
Example 11. If A + B + C = θ and cotθ = cot A + cot B + cot C, show that , sin(Aθ). sin (Bθ).sin (Cθ)= sin^{3} θ.
Solution:
Given cot θ = cot A + cot B + cot C or cot θ  cot A = cot B + cot C
....(1)
....(2)
....(3)
Multiplying (1) , (2) and (3) we get the result
Example 12. Find whether a triangle ABC can exist with the tangents of its interior angle satisfying, tan A = x, tan B = x + 1, and tan C = 1  x for some real value of x. Justify your assertion with adequate reasoning.
Solution:
In a triangle ∑ tan A = π tan A (to be proved)
x + x + 1 + 1 – x = x(1 + x)(1 – x)
2 + x = x – x^{3}; x^{3} = –2, x = 2^{1/3}
Hence tanA = x < 0 and tanB = x + 1 = 1 – 2^{1/3} < 0
Hence A and B both are obtuse. Which is not possible in a triangle. Hence no such triangle can exist.
Example 13._{ }Find the greatest value of c such that system of equations x^{2} + y^{2} = 25; x + y = c has a real solution.
Solution:
Put x = 5 cosθ and y = 5 sinθ
5(cosθ+ sinθ) = c; but (cosθ + sinθ)max = √2 and (cosθ + sinθ)min = – √2
hence, c_{max} = 5√2
Example 14. Find the minimum and maximum value of f (x, y) = 7x^{2} + 4xy + 3y^{2} subjected to x^{2} + y^{2} = 1.
Solution:
Let x = cosθ and y = sinθ
y = f (θ) = 7 cos2θ + 4 sin θcosθ + 3 sin2θ = 3 + 2 sin 2q + 2(1 + cos 2θ)
= 5 + 2(sin 2θ + cos 2θ) but √2 ≤ (sin 2θ + cos 2θ) ≤ √2
y_{max} = 5 + 2√2 and y_{min} = 5 – 2√2
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1. What are the basic trigonometric identities? 
2. How do you find the trigonometric functions of allied angles? 
3. What are some examples of multiple angles in trigonometry? 
4. How do you determine the maximum and minimum values of trigonometric functions? 
5. What are some important conditional identities in trigonometry? 

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