sin θ as `"opp"/"hyp"`; cos θ as `"adj"/"hyp"`, and. tan θ as `"opp"/"adj"`, but we are using the specific x-, y- and r-values defined by the point (x, y) that the terminal side passes through. We can choose any point on that line, of course, to define our ratios. To find r, we use Pythagoras' Theorem, since we have a right angled triangle:

Trigonometric functions and their reciprocals on the unit circle. All of the right-angled triangles are similar, i.e. the ratios between their corresponding sides are the same. For sin, cos and tan the unit-length radius forms the hypotenuse of the triangle that defines them.

The value of sin θ is the y-coördinate of the endpoint of the unit radius. The value of cos θ is the x -coördinate. The following shows the coördinates of the endpoint on the unit circle of every quadrantal angle :

For the `tan theta` case, y is positive and x is negative, so `y/x` will always be negative. Considering the other quadrants, we see a pattern. In Quadrant II, `sin theta` is positive, `cos theta` and `tan theta` are negative.

In any triangle we have: 1 - The sine law sin A / a = sin B / b = sin C / c 2 - The cosine laws a 2 = b 2 + c 2 - 2 b c cos A b 2 = a 2 + c 2 - 2 a c cos B c 2 = a 2 + b 2 - 2 a b cos C Relations Between Trigonometric Functions

The point of the unit circle is that it makes other parts of the mathematics easier and neater. For instance, in the unit circle, for any angle θ, the trig values for sine and cosine are clearly nothing more than sin(θ) = y and cos(θ) = x.

The law of sines states that for an arbitrary triangle with sides a, b, and c and angles opposite those sides A, B and C: ⁡ = ⁡ = ⁡ =, where Δ is the area of the triangle, or, equivalently, ⁡ = ⁡ = ⁡ =, where R is the triangle's circumradius.

Dec 16, 2024 · If sin θ = x Then putting sin on the right side θ = sin -1 x sin -1 x = θ So, inverse of sin is an angle. Similarly, inverse of all the trigonometry function is angle. Note : Here angle is measured in radians, not degrees. So, we have sin -1 x cos -1 x tan -1 x cosec -1 x sec -1 x tan -1 x Domain and Range of Inverse Trigonometric Functions

Jul 2, 2022 · The sine function: \(\sin (\theta )=\dfrac{y}{r}\) The cosine function: \(\cos (\theta )=\dfrac{x}{r}\)

sin θ, The ratios x/r and y/r are well defined provided r ≠ 0. Recall that r measures the distance from the point (x, y) ≠ (0, 0) lying on the terminal side of an angle in standard position to the origin. By definition, r cannot be zero. Thus, all values of θ are valid inputs.