Trig functions and key identities

Trigonometry connects angles and ratios

Trigonometry is the mathematics of angles. In physics, trig functions appear whenever direction matters: forces on ramps, projectile motion, wave motion, circular motion, optics, and rotations.

For a right triangle with angle $heta$, the basic definitions are

sin heta = rac{opposite}{hypotenuse}

cos heta = rac{adjacent}{hypotenuse}

an heta = rac{opposite}{adjacent}

These ratios depend only on the angle, not the size of the triangle. That is why trig functions are so useful.

Components of a vector

Suppose a vector has magnitude $A$ and makes an angle $heta$ with the positive x-axis. Its components are

$A_x = Acos heta$

$A_y = Asin heta$

This pattern appears constantly. If a force pulls at an angle, its horizontal and vertical effects are found using cosine and sine. If a projectile is launched at speed $v_0$ and angle $heta$, its initial velocity components are $v_0cos heta$ and $v_0sin heta$.

The exact use of sine and cosine depends on how the angle is measured. Always draw a diagram.

The unit circle

The right-triangle definitions work naturally for acute angles, but physics needs angles beyond $90^circ$. The unit circle extends trig functions to all angles. On a circle of radius 1, a point at angle $heta$ has coordinates

$(cos heta,sin heta)$

This means cosine is the x-coordinate and sine is the y-coordinate. As the point moves around the circle, sine and cosine repeat. This periodic behavior is why trig functions describe waves and oscillations.

Important values

Several angles appear often:

$sin 0 = 0,quad cos 0 = 1$

ight)=1,quad cosleft(rac{pi}{2} ight)=0$$ $$sin pi = 0,quad cos pi = -1$$ These values become easier to remember using the unit circle. ## The Pythagorean identity The most important trig identity is $$sin^2 heta + cos^2 heta = 1$$ This comes from the Pythagorean theorem applied to the unit circle. It is used throughout physics to simplify expressions and prove relationships. For example, if a velocity has components $vcos heta$ and $vsin heta$, then the magnitude is $$sqrt{(vcos heta)^2 + (vsin heta)^2} = v$$ because $cos^2 heta + sin^2 heta = 1$. ## Tangent and slope Tangent relates to slope. For a line making angle $ heta$ with the x-axis, $$ an heta = rac{rise}{run}$$ This is why tangent appears in ramps, inclines, and projectile angles. If you know the vertical and horizontal components of a vector, the angle can be found from $$ an heta = rac{A_y}{A_x}$$ with quadrant care. ## Periodicity and symmetry Sine and cosine repeat every $2pi$: $$sin( heta + 2pi) = sin heta$$ $$cos( heta + 2pi) = cos heta$$ They also have symmetry. Sine is odd, meaning $sin(- heta) = -sin heta$. Cosine is even, meaning $cos(- heta)=cos heta$. ## The big idea Trig functions translate between angles and components. The unit circle extends them to all angles and explains their repeating behavior. The identities are not arbitrary tricks; they express geometric facts. In physics, sine, cosine, and tangent are tools for turning direction into calculation.