6.2 Trigonometry of Right Triangles
6.2.1 Learning Objectives
By the end of this section, you should be able to:
- define trigonometric ratios in right triangles;
- find missing sides and acute angles;
- solve right triangles;
- use angles of elevation and depression; and
- model practical measurement problems.
6.2.2 Right-Triangle Ratios
For an acute angle \(\theta\),
\[ \sin\theta=\frac{\text{opposite}}{\text{hypotenuse}},\quad \cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}},\quad \tan\theta=\frac{\text{opposite}}{\text{adjacent}}. \]
The reciprocal ratios define cosecant, secant, and cotangent.
Example 6.5 Finding Trigonometric Ratios
A right triangle has sides \(5\), \(12\), and \(13\). For the angle opposite side \(5\), find sine, cosine, and tangent.
Solution
\[ \boxed{\sin\theta=5/13,\quad \cos\theta=12/13,\quad \tan\theta=5/12} \]
Exercise 6.5 For the angle opposite side \(12\), find \(\sin\theta\) in the same triangle.
Show answer
Answer: \(\boxed{12/13}\)
6.2.3 Solving Right Triangles
Choose a ratio containing the known and unknown sides. To find an angle, use an inverse trigonometric function. The two acute angles are complementary.
Example 6.6 Finding a Missing Side and Angle
A right triangle has hypotenuse \(18\text{ cm}\) and an acute angle of \(35^\circ\). Find the opposite side and the other acute angle.
Solution
\[ x=18\sin35^\circ\approx10.32\text{ cm}, \]
\[ 90^\circ-35^\circ=55^\circ. \]
\[ \boxed{x\approx10.32\text{ cm};\quad 55^\circ} \]
Exercise 6.6 A right triangle has opposite side \(7\) and adjacent side \(10\). Find the angle to the nearest tenth of a degree.
Show answer
Answer: \(\boxed{\theta=\tan^{-1}(7/10)\approx35.0^\circ}\)
6.2.4 Angles of Elevation and Depression
An angle of elevation is measured upward from a horizontal line. An angle of depression is measured downward. Parallel horizontal lines often create equal alternate interior angles.
Example 6.7 Finding a Building Height
From a point \(40\text{ m}\) from a building, the angle of elevation to the roof is \(38^\circ\). Find the height, assuming level ground.
Solution
\[ \tan38^\circ=\frac h{40}, \qquad h=40\tan38^\circ\approx31.25. \]
\[ \boxed{h\approx31.25\text{ m}} \]
Exercise 6.7 A kite is observed at an elevation angle of \(52^\circ\) from \(25\text{ m}\) away horizontally. Find its height.
Show answer
Answer: \(\boxed{25\tan52^\circ\approx32.00\text{ m}}\)
6.2.5 Conceptual Takeaways
- Trigonometric ratios compare side lengths relative to an angle.
- The hypotenuse is always opposite the right angle.
- Inverse trigonometric functions recover angles from ratios.
- A diagram connects a practical situation to a right triangle.
- Units and rounding should match the context.
6.2.6 Skills You Should Be Able to Do
- Label opposite, adjacent, and hypotenuse sides.
- Choose and apply an appropriate ratio.
- Find missing sides and acute angles.
- Solve an entire right triangle.
- Model elevation and depression problems.
6.2.7 Practice Problems with Solutions
In a right triangle, \(\theta=28^\circ\) and the hypotenuse is \(15\). Find the opposite side.
Show Solution
\[ x=15\sin28^\circ\approx\boxed{7.04}. \]
Find \(\theta\) if the opposite side is \(9\) and adjacent side is \(14\).
Show Solution
\[ \theta=\tan^{-1}(9/14)\approx\boxed{32.7^\circ}. \]
A right triangle has legs \(8\) and \(15\). Find its hypotenuse.
Show Solution
\[ c=\sqrt{8^2+15^2}=\boxed{17}. \]
A \(6\text{ m}\) ladder makes a \(68^\circ\) angle with level ground. How high does it reach?
Show Solution
\[ h=6\sin68^\circ\approx\boxed{5.56\text{ m}}. \]
From \(55\text{ m}\) away, a tower has elevation angle \(41^\circ\). Find its height.
Show Solution
\[ h=55\tan41^\circ\approx\boxed{47.81\text{ m}}. \]
A right triangle has hypotenuse \(20\) and one leg \(12\). Find both acute angles.
Show Solution
\[ \theta=\sin^{-1}(12/20)\approx36.9^\circ. \]
The other angle is \(90^\circ-36.9^\circ=53.1^\circ\).
\[ \boxed{36.9^\circ\text{ and }53.1^\circ} \]