Trigonometry is an important branch of mathematics, it deals with the relationship of sides with angles in a right-angle triangle. Trigonometry is important in physics as well, it is used to discover, the height of towers, the altitude betwixt stars, or in the Navigation systems. Trigonometry is predicated on the principle that “If two triangles have an equivalent set of angles then their sides are within the aforementioned ratio”. Side lengths are oftentimes different but side ratios are equivalent.
Trigonometric functions or circular functions or trigonometric ratios show
The relationship of angles and sides.
These trigonometric ratios are obtained past taking ratios of sides. We have six trigonometric ratios Sin, Cos, Tan, Cosec, Sec, Cot.
- sin A = Perpendicular / Hypotenuse
- cos A = Base / Hypotenuse
- tan A = Perpendicular / Base
- cot A = Base / Perpendicular
- sec A = Hypotenuse / Base
- cosec A = Hypotenuse / Perpendicular.
Here, A is that the angle reverse to the perpendicular side. Let’s see what the Perpendicular, base, and hypotenuse of a right triangle are,
The side ahead of the angle is perpendicular. In this example, the side in forepart of 30 deg is called information technology’s perpendicular.
A base is ane amidst the edges which contain an angle, except hypotenuse.
It is a side opposite to 90°. It is the largest side.
Perpendicular and base changes equally bending changes. During a triangle, a side is perpendicular for an bending, but an equivalent side may exist a base for an additional angle, but the hypotenuse remains an equivalent considering it’s a side opposite to angle 90°.
As shown inside the to a higher place diagram for an equivalent triangle if considered angle 30° the perpendicular is that the side PQ, merely if considered angle 60° the perpendicular is the side QR.
What is all sin tan cos rule?
All sin tan cos rule tells about the sign(+ve or -ve) of trigonometric ratios. When trigonometric ratios are calculated of all angles betwixt 0° to 360° or 0° to 2π we also have to take intendance of the sign(+ve or -ve) of Sine, Cos, and tan. Since the other 3 trigonometric ratios are reciprocal of these iii their sign is decided by these iii ratios.
All sin tan cos rule says that,
- When angle θ lies in the first quadrant 0° ≤ θ ≤ xc°
all three ratios
i.east. Sine, cos, and tan are positive with their reciprocals.
- When bending θ lies in the 2d quadrant i.e. 90° ≤ θ ≤ 180°
sin and its reciprocal cosec
are positive and other ratios tan and cos are negative in the second quadrant.
- When angle θ lies in the third quadrant i.e. 180° ≤ θ ≤ 270°
tan and its reciprocal cot
are positive and other ratios sin and cos are negative in the third quadrant.
- When angle θ lies in the 4th quadrant i.e. 270° ≤ θ ≤ 360°
cos and its reciprocal sec
are positive and other ratios sin and tan are negative in the 4th quadrant.
Question1. Find the sign of Sin(150°), Sin(255°), Cos(160°), Cos(320°), tan(170°), tan (350°).
Co-ordinate to all sin tan cos rule,
Sin(150°) = +ve
Sin(225°) = -ve
Cos(160°) = -ve
Cos(320°) = +ve
tan(170°) = +ve
tan(350°) = -ve
Question2: find the value of sin(225°), tan(225°), cos(225°).
225° lies in the 3rd quadrant, According to all sin tan cos dominion in 3rd quadrant tan is positive and sin and cos are negative.
sin(225°) = sin(180° + 45°)
Sin(225°) = -one/√2
tan(225°) = sin(180° + 45°)
Tan(225°) = 1
Cos(225°) = Cos(180° + 45°)
cos(225°) = -i/√2
Solution 3: Observe Value of Sin(225°) – Cos(315°).
Sin(225°) = -one/√2
Cos(315°) = 1/√ii
-1/√2 – 1/√two = – √two
Sin(225°) – Cos(315°) = -√2