The following discussion is limited to vectors in a two‐dimensional coordinate plane, although the concepts can be extended to higher dimensions. If vector is shifted so that its initial point is at the origin of the rectangular coordinate plane, it is said to be in standard position. If vector is equal to vector and has its initial point at… Continue reading Rectangular Coordinate System
Month: September 2022
Pythagorean Identities
sin2θ+cos2θ=1sin2θ+cos2θ=1 cot2θ+1=cosec2θcot2θ+1=cosec2θ 1+tan2θ=sec2θ
Inverse Trigonometry Formulas
sin-1 (–x) = – sin-1 x cos-1 (–x) = π – cos-1 x tan-1 (–x) = – tan-1 x cosec-1 (–x) = – cosec-1 x sec-1 (–x) = π – sec-1 x cot-1 (–x) = π – cot-1 x
Sum to Product Formula List
We can prove these sum to product formulas using the product to sum formulas in trigonometry. The sum to product formula are expressed as follows: sin A + sin B = 2 sin [(A + B)/2] cos [(A – B)/2] sin A – sin B = 2 sin [(A – B)/2] cos [(A + B)/2] cos A… Continue reading Sum to Product Formula List
Half Angle Formula of Tan Derivation
We know that tan (A/2) = [sin (A/2)] / [cos (A/2)] From the half angle formulas of sin and cos, tan (A/2) = [±√(1 – cos A)/2] / [±√(1 + cos A)/2] = ±√[(1 – cos A) / (1 + cos A)] This is one of the formulas of tan (A/2). Let us derive the other two… Continue reading Half Angle Formula of Tan Derivation
Half Angle Formula of Cos Derivation
Now, we will prove the half angle formula for the cosine function. Using one of the above formulas of cos A, cos A = 2 cos2(A/2) – 1 From this, 2 cos2(A/2) = 1 + cos A cos2 (A/2) = (1 + cos A) / 2 cos (A/2) = ±√[(1 + cos A) / 2]
Half Angle Formula of Sin Proof
Now, we will prove the half angle formula for the sine function. Using one of the above formulas of cos A, we have cos A = 1 – 2 sin2 (A/2) From this, 2 sin2 (A/2) = 1 – cos A sin2 (A/2) = (1 – cos A) / 2 sin (A/2) = ±√[(1 – cos A) /… Continue reading Half Angle Formula of Sin Proof
Half Angle Formulas?
In this section, we will see the half angle formulas of sin, cos, and tan. We know the values of the trigonometric functions (sin, cos , tan, cot, sec, cosec) for the angles like 0°, 30°, 45°, 60°, and 90° from the trigonometric table. But to know the exact values of sin 22.5°, tan 15°, etc,… Continue reading Half Angle Formulas?
Triple Angle Identities
Sin 3x = 3sin x – 4sin3x Cos 3x = 4cos3x-3cos x T a n 3 x = 3 t a n x − t a n 3 x / 1 − 3 t a n 2 x
Double Angle Formulas of Tan
The sum formula of tangent function is, tan (A + B) = (tan A + tan B) / (1 – tan A tan B) When A = B, the above formula becomes, tan (A + A) = (tan A + tan A) / (1 – tan A tan A) =(2 tan A) / (1 – tan2A)… Continue reading Double Angle Formulas of Tan