Trigonometric Identities Made Simple: Functions, Angles, and Meaning

1) The Three Core Trig Functions (Right-Triangle View)

Start with a right triangle and pick one acute angle \(\theta\). Label the side opposite the angle as opposite, the side next to the angle (but not the hypotenuse) as adjacent, and the longest side as hypotenuse.

• Sine: \(\sin\theta=\dfrac{\text{opposite}}{\text{hypotenuse}}\)
• Cosine: \(\cos\theta=\dfrac{\text{adjacent}}{\text{hypotenuse}}\)
• Tangent: \(\tan\theta=\dfrac{\text{opposite}}{\text{adjacent}}\)

These ratios depend only on the angle, not on the triangle’s size. If you scale the triangle up or down, all side lengths scale by the same factor, so the ratios stay the same.

Quick guided check (triangle ratios)

Suppose a right triangle has sides (relative to angle \(\theta\)): opposite = 3, adjacent = 4, hypotenuse = 5. Compute:

• \(\sin\theta=3/5\)
• \(\cos\theta=4/5\)
• \(\tan\theta=3/4\)

Now imagine the triangle is doubled: opposite = 6, adjacent = 8, hypotenuse = 10. Recompute the ratios and confirm they match.

2) Extending to All Angles (Unit Circle View)

Right-triangle definitions only cover acute angles (\(0^\circ<\theta<90^\circ\)). To define trig functions for any angle (including obtuse, negative, and angles beyond one full turn), use the unit circle.

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The unit circle is the circle of radius 1 centered at the origin. For an angle \(\theta\) in standard position (starting on the positive x-axis and rotating), the point where the terminal side meets the unit circle is \((x,y)\). Then:

• \(\cos\theta=x\)
• \(\sin\theta=y\)
• \(\tan\theta=\dfrac{y}{x}\) (when \(x\neq 0\))

This matches the triangle ratios for acute angles: dropping a perpendicular to the x-axis forms a right triangle with hypotenuse 1, adjacent =x, opposite =y.

Practical step-by-step: reading trig values from a unit-circle point

1. Identify the point \((x,y)\) on the unit circle.
2. Set \(\cos\theta=x\) and \(\sin\theta=y\).
3. If needed, compute \(\tan\theta=y/x\) (only if \(x\neq 0\)).

Example: If the terminal point is \((\tfrac{1}{2},\tfrac{\sqrt{3}}{2})\), then \(\cos\theta=\tfrac{1}{2}\), \(\sin\theta=\tfrac{\sqrt{3}}{2}\), and \(\tan\theta=\sqrt{3}\).

3) Angle Measure: Degrees vs Radians

Angles can be measured in degrees or radians. Degrees split a full rotation into 360 equal parts. Radians measure angle by arc length: on a circle of radius \(r\), an angle of \(\theta\) radians cuts off arc length \(s=r\theta\). On the unit circle (\(r=1\)), arc length equals the radian measure.

Conversion rules:

• \(\text{radians}=\text{degrees}\cdot\dfrac{\pi}{180}\)
• \(\text{degrees}=\text{radians}\cdot\dfrac{180}{\pi}\)

Quick guided check (conversion)

Convert \(60^\circ\) to radians: \(60\cdot\pi/180=\pi/3\). Convert \(5\pi/6\) to degrees: \((5\pi/6)\cdot 180/\pi=150^\circ\).

4) Signs by Quadrant (Why Sine/Cosine Can Be Negative)

On the unit circle, \(\cos\theta\) is the x-coordinate and \(\sin\theta\) is the y-coordinate. So their signs come from whether x and y are positive or negative in each quadrant.

Practical step-by-step: predicting signs without calculating exact values

1. Locate the quadrant of \(\theta\) (for example, \(120^\circ\) is in Quadrant II).
2. Use the quadrant sign table to decide the signs of \(\sin\theta\), \(\cos\theta\), and \(\tan\theta\).

Example: \(\theta=300^\circ\) is Quadrant IV, so \(\sin\theta<0\), \(\cos\theta>0\), and \(\tan\theta<0\).

5) What an Identity Is (and What It Is Not)

An identity is a statement that is true for all angles where both sides are defined. Think of it as a permanent rule of the trig system.

An equation to solve is a statement that is true only for certain angles, and your job is to find those angles.

Identity vs equation (side-by-side)

• Identity example: \(\sin^2\theta+\cos^2\theta=1\) (true for every real \(\theta\))
• Equation-to-solve example: \(\sin\theta=\tfrac{1}{2}\) (true only for specific angles)

Also note the phrase “where defined.” For instance, \(\tan\theta=\dfrac{\sin\theta}{\cos\theta\) is an identity, but only for angles with \(\cos\theta\neq 0\) (because division by zero is not allowed).

6) Guided Checks: Verifying Statements by Substitution

One of the fastest ways to build intuition is to test a proposed identity at a few angles you know well. This does not replace a proof, but it is an excellent error-check and meaning-check.

Check A: \(\sin^2\theta+\cos^2\theta=1\)

Pick angles with easy unit-circle coordinates.

• At \(\theta=0\): \(\sin 0=0\), \(\cos 0=1\). Then \(0^2+1^2=1\).
• At \(\theta=\pi/2\): \(\sin(\pi/2)=1\), \(\cos(\pi/2)=0\). Then \(1^2+0^2=1\).
• At \(\theta=\pi\): \(\sin\pi=0\), \(\cos\pi=-1\). Then \(0^2+(-1)^2=1\).

Why this identity makes sense: on the unit circle, the point \((\cos\theta,\sin\theta)\) always satisfies \(x^2+y^2=1\). Substituting x=\cos\theta and y=\sin\theta gives \(\cos^2\theta+\sin^2\theta=1\).

Check B: \(\tan\theta=\dfrac{\sin\theta}{\cos\theta}\) (where \(\cos\theta\neq 0\))

• At \(\theta=\pi/4\): \(\sin\theta=\cos\theta=\tfrac{\sqrt{2}}{2}\). Then \(\sin\theta/\cos\theta=1\), and \(\tan\theta=1\).
• At \(\theta=\pi\): \(\sin\pi=0\), \(\cos\pi=-1\). Then \(\sin\pi/\cos\pi=0\), and \(\tan\pi=0\).

Also check a “not defined” case: at \(\theta=\pi/2\), \(\cos\theta=0\), so \(\sin\theta/\cos\theta\) is undefined, matching the fact that \(\tan(\pi/2)\) is undefined.

Check C: A statement that is not an identity

Consider \(\sin\theta=\cos\theta\). Test two angles:

• At \(\theta=\pi/4\): \(\sin\theta=\cos\theta\) (true).
• At \(\theta=0\): \(\sin 0=0\), \(\cos 0=1\) (false).

So \(\sin\theta=\cos\theta\) is not an identity; it is an equation that is true only for certain angles (and those angles are what you would solve for).

7) Meaning First: How Identities “Work” Conceptually

Many identities are really geometry or coordinate facts written in trig language. Two recurring meaning anchors are:

• Coordinates: \((\cos\theta,\sin\theta)\) is a point on the unit circle.
• Ratios: \(\tan\theta\) compares vertical change to horizontal change via \(y/x\).

When you see an identity, try asking: “Is this a restatement of a circle fact, a ratio fact, or a sign/quadrant fact?” Then do a quick substitution check with a couple of angles to confirm you are reading it correctly.