The Main Idea

Right triangles give us a straightforward way to define and evaluate trigonometric functions. By labeling the sides relative to an acute angle [latex]\theta[/latex]—opposite, adjacent, and hypotenuse—we can express sine, cosine, tangent, and their reciprocals as ratios of side lengths. This geometric perspective is often the first way students encounter trigonometry and is especially useful for evaluating trig functions of special angles like [latex]30^\circ[/latex], [latex]45^\circ[/latex], and [latex]60^\circ[/latex].

Quick Tips: Using Right Triangles

1. Recall the Definitions

   • [latex]\sin\theta = \dfrac{\text{opposite}}{\text{hypotenuse}}[/latex]

   • [latex]\cos\theta = \dfrac{\text{adjacent}}{\text{hypotenuse}}[/latex]

   • [latex]\tan\theta = \dfrac{\text{opposite}}{\text{adjacent}}[/latex]

   • [latex]\csc\theta = \dfrac{\text{hypotenuse}}{\text{opposite}}[/latex]

   • [latex]\sec\theta = \dfrac{\text{hypotenuse}}{\text{adjacent}}[/latex]

   • [latex]\cot\theta = \dfrac{\text{adjacent}}{\text{opposite}}[/latex]

2. Use the Pythagorean Theorem

   • If two sides are known, find the third: [latex]a^{2} + b^{2} = c^{2}[/latex].

3. Special Right Triangles Help

   • [latex]45^\circ-45^\circ-90^\circ[/latex] triangle: legs = 1, hypotenuse = [latex]\sqrt{2}[/latex].

   • [latex]30^\circ-60^\circ-90^\circ[/latex] triangle: shorter leg = 1, longer leg = [latex]\sqrt{3}[/latex], hypotenuse = 2.

   • Use these ratios to evaluate trig functions of [latex]30^\circ[/latex], [latex]45^\circ[/latex], and [latex]60^\circ[/latex].

4. Remember the Context

   • Right-triangle definitions only apply to acute angles inside right triangles, but they extend naturally to the unit circle for all angles.

The Main Idea

Cofunction identities show that the trig function of an angle equals the cofunction of its complement. Since two acute angles in a right triangle add up to [latex]90^\circ[/latex] (or [latex]\dfrac{\pi}{2}[/latex]), the sine of one angle equals the cosine of the other, tangent pairs with cotangent, and secant pairs with cosecant. These relationships are useful for simplifying expressions and recognizing equivalences between trig values.

The cofunction identities are:

• [latex]\sin\theta = \cos\left(90^\circ - \theta\right)[/latex] or [latex]\sin\theta = \cos\left(\dfrac{\pi}{2}-\theta\right)[/latex]

• [latex]\cos\theta = \sin\left(90^\circ - \theta\right)[/latex]

• [latex]\tan\theta = \cot\left(90^\circ - \theta\right)[/latex]

• [latex]\cot\theta = \tan\left(90^\circ - \theta\right)[/latex]

• [latex]\sec\theta = \csc\left(90^\circ - \theta\right)[/latex]

• [latex]\csc\theta = \sec\left(90^\circ - \theta\right)[/latex]

Quick Tips: Using Cofunctions

1. Think Right Triangle

   • In a right triangle, if one acute angle is [latex]\theta[/latex], the other is [latex]90^\circ - \theta[/latex].

   • The sine of one equals the cosine of the other, and so on.

2. Memorize the Pairs

   • Sine ↔ Cosine

   • Tangent ↔ Cotangent

   • Secant ↔ Cosecant

The Main Idea

The definitions of trigonometric functions extend beyond right triangles to any angle by using the unit circle or coordinates in the Cartesian plane. For an angle [latex]\theta[/latex] drawn in standard position with a point [latex](x,y)[/latex] on its terminal side and distance [latex]r=\sqrt{x^{2}+y^{2}}[/latex], the trig functions are defined as:

• [latex]\sin\theta = \dfrac{y}{r}[/latex]

• [latex]\cos\theta = \dfrac{x}{r}[/latex]

• [latex]\tan\theta = \dfrac{y}{x}, \quad x \neq 0[/latex]

• [latex]\csc\theta = \dfrac{r}{y}, \quad y \neq 0[/latex]

• [latex]\sec\theta = \dfrac{r}{x}, \quad x \neq 0[/latex]

• [latex]\cot\theta = \dfrac{x}{y}, \quad y \neq 0[/latex]

This framework works for all real angles, not just acute ones, and naturally incorporates negative values and quadrants.

Quick Tips: Using Definitions for Any Angle

1. Unit Circle Connection

   • On the unit circle, [latex]r=1[/latex], so [latex]\sin\theta=y[/latex] and [latex]\cos\theta=x[/latex].

2. Signs Depend on Quadrant

   • Quadrant I: all functions positive.

   • Quadrant II: sine and cosecant positive.

   • Quadrant III: tangent and cotangent positive.

   • Quadrant IV: cosine and secant positive.

3. Always Reduce to Ratios

   • No matter the quadrant, definitions reduce trig to [latex]\dfrac{x}{r}[/latex] and [latex]\dfrac{y}{r}[/latex], with signs handled automatically.

The Main Idea

Right triangle trigonometry helps us solve real-world problems that involve heights, distances, and angles. By modeling a situation as a right triangle, we can use sine, cosine, tangent, and their reciprocals to connect side lengths with angles. These tools are especially useful for applications such as measuring building heights, finding the distance across rivers, or calculating the angle of elevation or depression.

Quick Tips: Solving Applied Problems

1. Draw a Diagram

   • Sketch the scenario as a right triangle.

   • Label the known sides and angles, and mark the unknown quantity.

2. Identify the Function You Need

   • Use sine if you know opposite and hypotenuse.

   • Use cosine if you know adjacent and hypotenuse.

   • Use tangent if you know opposite and adjacent.

3. Angle of Elevation/Depression

   • Elevation: angle measured upward from horizontal.

   • Depression: angle measured downward from horizontal.

   • Both are drawn with respect to a horizontal line.

4. Use Inverse Trig When Needed

   • If side lengths are known but the angle is unknown, use [latex]\sin^{-1}[/latex], [latex]\cos^{-1}[/latex], or [latex]\tan^{-1}[/latex].

5. Check Units and Context

   • Make sure answers are expressed with appropriate units (feet, meters, degrees, etc.).

   • Interpret the solution in the context of the problem, not just as a number.