7.2 Right Triangle Trigonometry

Right triangle trigonometry unlocks the power to solve real-world problems using simple ratios. By understanding sine, cosine, and tangent, you can find missing sides and angles in triangles, opening doors to applications in surveying, navigation, and more.

Memorizing common angle values and using complementary angles simplifies calculations. The unit circle extends these concepts beyond right triangles, allowing you to work with any angle in the coordinate plane. This foundation sets the stage for advanced trigonometric concepts.

Right Triangle Trigonometry

Trigonometric functions in right triangles

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• Find missing side lengths and angles in right triangles using trigonometric functions sine (sin), cosine (cos), and tangent (tan)
  • $\sin \theta = \frac{\text{[opposite](https://www.fiveableKeyTerm:Opposite)}}{\text{[hypotenuse](https://www.fiveableKeyTerm:Hypotenuse)}}$ ratio of opposite side length to hypotenuse length
  • $\cos \theta = \frac{\text{[adjacent](https://www.fiveableKeyTerm:Adjacent)}}{\text{hypotenuse}}$ ratio of adjacent side length to hypotenuse length
  • $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$ ratio of opposite side length to adjacent side length
• Inverse trigonometric functions $\sin^{-1}$, $\cos^{-1}$, and $\tan^{-1}$ used to find angle measures from side length ratios
  • $\sin^{-1}(\frac{\text{opposite}}{\text{hypotenuse}})$ finds angle $\theta$ given opposite and hypotenuse lengths
  • $\cos^{-1}(\frac{\text{adjacent}}{\text{hypotenuse}})$ finds angle $\theta$ given adjacent and hypotenuse lengths
  • $\tan^{-1}(\frac{\text{opposite}}{\text{adjacent}})$ finds angle $\theta$ given opposite and adjacent lengths
• These ratios, known as trigonometric ratios, are derived from the properties of similar triangles

Common angle trigonometric values

• 30-60-90 triangles have side length ratios $x:\sqrt{3}x:2x$ and angle measures 30°, 60°, and 90°
  • $\sin 30° = \frac{1}{2}$, $\cos 30° = \frac{\sqrt{3}}{2}$, $\tan 30° = \frac{\sqrt{3}}{3}$
  • $\sin 60° = \frac{\sqrt{3}}{2}$, $\cos 60° = \frac{1}{2}$, $\tan 60° = \sqrt{3}$
• 45-45-90 triangles have side length ratios $x:x:\sqrt{2}x$ and angle measures 45°, 45°, and 90°
  • $\sin 45° = \frac{\sqrt{2}}{2}$, $\cos 45° = \frac{\sqrt{2}}{2}$, $\tan 45° = 1$
• Memorize common angle trigonometric values to quickly solve problems without calculations

Complementary angles and cofunctions

• Complementary angles sum to 90° ($\alpha + \beta = 90°$)
• Cofunction identities relate trigonometric functions of complementary angles
  • $\sin \alpha = \cos \beta$ and $\cos \alpha = \sin \beta$
  • $\tan \alpha = \cot \beta$ and $\cot \alpha = \tan \beta$
  • $\sec \alpha = \csc \beta$ and $\csc \alpha = \sec \beta$
• Use cofunction identities to simplify trigonometric expressions and solve equations

Trigonometric functions for any angle

• Unit circle defines trigonometric functions for any angle $\theta$
  • $\sin \theta = \frac{y}{r}$, $\cos \theta = \frac{x}{r}$, where $(x,y)$ is a point on the unit circle and $r$ is the radius
  • $\tan \theta = \frac{y}{x}$, $\cot \theta = \frac{x}{y}$, where $x \neq 0$ and $y \neq 0$
  • $\sec \theta = \frac{r}{x}$, $\csc \theta = \frac{r}{y}$, where $x \neq 0$ and $y \neq 0$
• Extend trigonometric functions beyond acute angles in right triangles to any angle measure in standard position

Quadrants and Angles of Reference

• The coordinate plane is divided into four quadrants, numbered counterclockwise from I to IV
• An angle of reference is the acute angle formed between the terminal side of an angle and the x-axis
• Trigonometric functions can be related to their values in the first quadrant using angles of reference

Applications of right-triangle trigonometry

1. Identify known and unknown values in the problem
2. Sketch a right triangle and label known and unknown sides and angles
3. Choose appropriate trigonometric function based on known and unknown values
4. Substitute known values into trigonometric function and solve for unknown value
5. Interpret result in context of problem and check reasonableness of answer

• Real-world applications include finding heights of buildings, distances between objects, and angles of elevation or depression (surveying, navigation)