Question

11. Write all the Trigonometric Equations ?​

Answer 1

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Basic Formulas

• sin θ = Opposite Side/Hypotenuse.
• Cos θ = Adjacent Side/Hypotenuse.
• tan θ = Opposite Side/Adjacent Side.
• sec θ = Hypotenuse/Adjacent Side.
• cosec θ = Hypotenuse/Opposite Side.
• cot θ = Adjacent Side/Opposite Side.

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Answer 2

Step-by-step explanation:

rigonometry ratios for a right-angled triangle can be written as;

sinθ = OppositesideHypotenuse

cosθ = AdjacentSideHypotenuse

tanθ = OppositesideAdjacentSide

secθ = HypotenuseAdjacentside

cosecθ = HypotenuseOppositeside

cotθ = AdjacentsideOppositeside

Trigonometric Ratios for Unit Circle

Similarly, for a unit circle, for which radius is equal to 1, and θ is the angle. The value of hypotenuse and adjacent side here is equal to the radius of the unit circle.

Hypotenuse = Adjacent side to θ = 1

Therefore, the ratios of trigonometry are given by:

sin θ = y/1 = y

cos θ = x/1 = x

tan θ = y/x

cot θ = x/y

sec θ = 1/x

cosec θ = 1/y

Trigonometry Identities

Tangent and Cotangent Identities

tanθ = sinθcosθ

cotθ = cosθsinθ

Reciprocal Identities

sinθ = 1/cosecθ

cosecθ = 1/sinθ

cosθ = 1/secθ

secθ = 1/cosθ

tanθ = 1/cotθ

cotθ = 1/tanθ

Pythagorean Identities

sin2θ + cos2θ = 1

1 + tan2θ = sec2θ

1 + cot2θ = cosec2θ

Even and Odd Angle Formulas

sin(-θ) = -sinθ

cos(-θ) = cosθ

tan(-θ) = -tanθ

cot(-θ) = -cotθ

sec(-θ) = secθ

cosec(-θ) = -cosecθ

Co-function Formulas

sin(900-θ) = cosθ

cos(900-θ) = sinθ

tan(900-θ) = cotθ

cot(900-θ) = tanθ

sec(900-θ) = cosecθ

cosec(900-θ) = secθ

Double Angle Formulas

sin2θ = 2 sinθ cosθ

cos2θ = 1 – 2sin2θ

tan2θ = 2tanθ1−tan2θ

Half Angle Formulas

sinθ = ±1−cos2θ2−−−−−−√

cosθ = ±1+cos2θ2−−−−−−√

tanθ = ±1−cos2θ1+cos2θ−−−−−−√

Thrice of Angle Formulas

sin3θ = 3sinθ – 4 sin3θ

Cos 3θ = 4cos3θ – 3 cosθ

Tan 3θ = 3tanθ–tan3θ1−3tan2θ

Cot 3θ = cot3θ–3cotθ3cot2θ−1

Sum and Difference Formulas

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) = TanA+TanB1–TanATanB

Tan (A-B) = TanA–TanB1+TanATanB

Product to Sum Formulas

Sin A Sin B = ½ [Cos (A-B) – Cos (A+B)]

Cos A Cos B = ½ [Cos (A-B) + Cos (A+B)]

Sin A Cos B = ½ [Sin (A+B) + Sin (A-B)]

Cos A Sin B = ½ [Sin (A+B) – Sin (A-B)]

Sum to Product Formulas

Sin A + Sin B = 2 sin A+B2 cos A−B2

Sin A – Sin B = 2 cosA+B2 sin A−B2

Cos A + Cos B = 2 cosA+B2 cos A−B2

Cos A – Cos B = – 2 sinA+B2 sin A−B2

Inverse Trigonometric Functions

If Sin θ = x, then θ = sin-1 x = arcsin(x)

Similarly,

θ = cos-1x = arccos(x)

θ = tan-1 x = arctan(x)

Also, the inverse properties could be defined as;

sin-1(sin θ) = θ

cos-1(cos θ) = θ

tan-1(tan θ) = θ

Unit Circle

With the help of unit circle, we can see here the different values of sin and cos ratios for different angles such as 0°, 30°, 45°, 60°, 90°, and so on in all the four quadrants.

Trigonometry Formulas List-Unit Circle

Trigonometry Table

Degrees 0° 30° 45° 60° 90° 180° 270° 360°

Radians 0 π/6 π/4 π/3 π/2 π 3π/2 2π

Sin θ 0 1/2 1/√2 √3/2 1 0 -1 0

Cos θ 1 √3/2 1/√2 1/2 0 -1 0 1

Tan θ 0 1/√3 1 √3 ∞ 0 ∞ 0

Cot θ ∞ √3 1 1/√3 0 ∞ 0 ∞

Sec θ 1 2/√3 √2 2 ∞ -1 ∞ 1

Cosec θ ∞ 2 √2 2/√3 1 ∞ -1 ∞

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