🔺Trigonometry Unit 2 – Acute Angles and Right Triangles

Key Concepts

• Acute angles measure less than 90 degrees
• Right triangles contain one 90-degree angle and two acute angles
• The side opposite the right angle is called the hypotenuse and is always the longest side
• The sides adjacent to the right angle are called legs or catheti
• Trigonometric ratios (sine, cosine, tangent) define relationships between the angles and sides of a right triangle
• Pythagorean theorem ($a^2 + b^2 = c^2$) relates the lengths of the three sides of a right triangle
• SOHCAHTOA is a mnemonic for remembering the trigonometric ratios (Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent)
• Inverse trigonometric functions (arcsin, arccos, arctan) help find angles when given side lengths

Angle Measurements

• Angles are measured in degrees (°) or radians (rad)
• One full rotation equals 360 degrees or $2\pi$ radians
• To convert from degrees to radians, multiply by $\frac{\pi}{180}$
  • For example, 45° = $\frac{45\pi}{180} = \frac{\pi}{4}$ radians
• To convert from radians to degrees, multiply by $\frac{180}{\pi}$
  • For example, $\frac{\pi}{6}$ radians = $\frac{\pi}{6} \times \frac{180}{\pi} = 30$°
• Acute angles are between 0° and 90° (or 0 and $\frac{\pi}{2}$ radians)
• Complementary angles add up to 90° (or $\frac{\pi}{2}$ radians)
• Supplementary angles add up to 180° (or $\pi$ radians)

Properties of Right Triangles

• Right triangles have one 90-degree angle
• The side opposite the right angle is called the hypotenuse (usually denoted as $c$)
• The other two sides are called legs or catheti (usually denoted as $a$ and $b$)
• The Pythagorean theorem states that in a right triangle, $a^2 + b^2 = c^2$
  • For example, if $a = 3$ and $b = 4$, then $c = \sqrt{3^2 + 4^2} = 5$
• The altitude (or height) of a right triangle is the perpendicular line segment from a vertex to the opposite side
• The median of a right triangle is a line segment from a vertex to the midpoint of the opposite side
• The angle bisector of a right triangle divides the opposite side into two segments proportional to the lengths of the other two sides

Trigonometric Ratios

• Sine (sin) of an angle is the ratio of the opposite side to the hypotenuse: $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$
• Cosine (cos) of an angle is the ratio of the adjacent side to the hypotenuse: $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$
• Tangent (tan) of an angle is the ratio of the opposite side to the adjacent side: $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$
• Reciprocal trigonometric ratios include cosecant (csc), secant (sec), and cotangent (cot)
  • $\csc \theta = \frac{1}{\sin \theta}$, $\sec \theta = \frac{1}{\cos \theta}$, $\cot \theta = \frac{1}{\tan \theta}$
• Trigonometric ratios are constant for a given angle, regardless of the triangle's size
• Special right triangles (30-60-90 and 45-45-90) have specific side length ratios that can be memorized

Solving Right Triangles

• To solve a right triangle, find the unknown side lengths or angle measures using given information
• Apply the Pythagorean theorem when given two side lengths to find the third side
  • For example, if $a = 5$ and $c = 13$, then $b = \sqrt{13^2 - 5^2} = 12$
• Use trigonometric ratios (SOHCAHTOA) when given one side length and one angle measure
  • For example, if $a = 8$ and $\theta = 30°$, then $c = \frac{a}{\cos \theta} = \frac{8}{\cos 30°} \approx 9.24$
• Inverse trigonometric functions (arcsin, arccos, arctan) help find angles when given side lengths
  • For example, if $a = 4$ and $c = 8$, then $\theta = \arcsin \frac{a}{c} = \arcsin \frac{4}{8} = 30°$
• Always double-check your solutions by verifying they satisfy the Pythagorean theorem and trigonometric ratios

Applications in Real Life

• Right triangles are used in various fields, such as architecture, engineering, and navigation
• In construction, right triangles help determine roof pitches, stair angles, and building heights
  • For example, a 30° roof pitch means the roof rises 30 vertical units for every 100 horizontal units
• Trigonometry is used in surveying to measure distances and angles between points
  • For instance, the height of a tree can be found by measuring the angle of elevation and the distance from the observer to the tree
• Navigation relies on right triangles to calculate distances and directions
  • For example, a plane flying at a 40° angle to the ground for 100 miles will have traveled about 64 miles horizontally (100 × cos 40°)
• Computer graphics and game development use trigonometry to rotate and transform objects in 2D and 3D space

Common Mistakes and How to Avoid Them

• Confusing sine, cosine, and tangent ratios
  • Remember SOHCAHTOA: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent
• Using the wrong angle or side when applying trigonometric ratios
  • Carefully identify the angle in question and its corresponding opposite, adjacent, and hypotenuse sides
• Forgetting to square or take the square root when using the Pythagorean theorem
  • The Pythagorean theorem is $a^2 + b^2 = c^2$, not $a + b = c$
• Mixing up degrees and radians
  • Always check the mode of your calculator and convert angles if necessary
• Rounding too early in multi-step problems
  • Carry extra decimal places throughout the calculation and round only the final answer
• Not checking if the triangle is a right triangle before applying right triangle trigonometry
  • Verify that one of the angles is 90° or that the side lengths satisfy the Pythagorean theorem

Practice Problems and Solutions

1. In a right triangle, if one of the acute angles is 35° and the hypotenuse is 20 units, find the lengths of the other two sides.

   • Solution:
     • Let the opposite side be $x$ and the adjacent side be $y$
     • $\sin 35° = \frac{x}{20}$, so $x = 20 \sin 35° \approx 11.47$ units
     • $\cos 35° = \frac{y}{20}$, so $y = 20 \cos 35° \approx 16.35$ units

2. A ladder 13 feet long leans against a wall. If the base of the ladder is 5 feet from the wall, find the angle the ladder makes with the ground.

   • Solution:
     • Let the angle be $\theta$
     • $\tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{12}{5}$
     • $\theta = \arctan \frac{12}{5} \approx 67.38°$

3. A ramp for wheelchairs must have a rise of no more than 1 unit for every 12 units of horizontal distance. What is the maximum angle the ramp can make with the ground?

   • Solution:
     • Let the angle be $\theta$
     • $\tan \theta = \frac{\text{rise}}{\text{run}} = \frac{1}{12}$
     • $\theta = \arctan \frac{1}{12} \approx 4.76°$

4. In a 30-60-90 triangle, if the shortest side is 6 units, find the lengths of the other two sides.

   • Solution:
     • In a 30-60-90 triangle, the sides are in the ratio of 1 : $\sqrt{3}$ : 2
     • If the shortest side (opposite to 30°) is 6 units, then:
       • The hypotenuse (opposite to 90°) is $6 \times 2 = 12$ units
       • The side opposite to 60° is $6 \times \sqrt{3} \approx 10.39$ units

5. Prove that in a 45-45-90 triangle, the length of the hypotenuse is $\sqrt{2}$ times the length of a leg.

   • Solution:
     • Let the length of each leg be $x$
     • By the Pythagorean theorem, $x^2 + x^2 = c^2$, where $c$ is the hypotenuse
     • Simplifying, $2x^2 = c^2$
     • Taking the square root of both sides, $\sqrt{2}x = c$
     • Therefore, the hypotenuse is $\sqrt{2}$ times the length of a leg