Mastering Right Triangles and Trigonometry for the SAT

Welcome to our comprehensive lesson on right triangles and trigonometry! This fundamental area of mathematics appears frequently on the SAT and forms the basis for many geometric and real-world applications. Whether you're starting from scratch or need a refresher, this lesson will guide you through all the essential concepts, formulas, and problem-solving techniques you'll need to excel on the SAT. We'll begin with the basics of right triangles, explore the Pythagorean theorem, and then dive into trigonometric ratios and their applications.

A right triangle is a triangle that has one angle measuring exactly 90 degrees (a right angle). The side opposite to the right angle is called the hypotenuse, which is always the longest side of the triangle. The other two sides are called the legs or catheti.

Right triangles have special properties that make them particularly useful in mathematics:

1. The Pythagorean Theorem: In a right triangle, the square of the length of the hypotenuse equals the sum of the squares of the lengths of the other two sides. If we label the sides as a, b (the legs) and c (the hypotenuse), then:

   $a^2+b^2=c^2$

2. Special Right Triangles:

   • 30°-60°-90° triangle: If the shortest leg has length $x$, then the hypotenuse has length $2x$, and the remaining leg has length $x\sqrt{3}$.
   • 45°-45°-90° triangle: Both legs have the same length $x$, and the hypotenuse has length $x\sqrt{2}$.

Trigonometry is the branch of mathematics that studies the relationships between the sides and angles of triangles. In the context of right triangles, trigonometry defines six fundamental ratios known as the trigonometric functions:

• Sine (sin): $\sin (\theta )=\frac{\text{opposite}}{\text{hypotenuse}}$
• Cosine (cos): $\cos (\theta )=\frac{\text{adjacent}}{\text{hypotenuse}}$
• Tangent (tan): $\tan (\theta )=\frac{\text{opposite}}{\text{adjacent}}=\frac{\sin (\theta )}{\cos (\theta )}$
• Cosecant (csc): $\csc (\theta )=\frac{1}{\sin (\theta )}=\frac{\text{hypotenuse}}{\text{opposite}}$
• Secant (sec): $\sec (\theta )=\frac{1}{\cos (\theta )}=\frac{\text{hypotenuse}}{\text{adjacent}}$
• Cotangent (cot): $\cot (\theta )=\frac{1}{\tan (\theta )}=\frac{\text{adjacent}}{\text{opposite}}$

On the SAT, you'll primarily work with sine, cosine, and tangent.

Step 1: Identify Right Triangles

Look for the right angle symbol (a small square in the corner) or information that tells you an angle is 90°. Sometimes, you'll need to create a right triangle by drawing an altitude or using the Pythagorean theorem to verify.

Step 2: Label the Parts Correctly

When working with a right triangle:

• Label the hypotenuse (the side opposite the right angle)
• Label the legs (the other two sides)
• When using trigonometry, identify which sides are "adjacent" and "opposite" relative to the angle you're working with

Step 3: Apply the Appropriate Formula

For finding missing sides:

• Use the Pythagorean theorem: $a^2+b^2=c^2$
• For special right triangles, use their specific ratios
• Use trigonometric ratios when angles are known

For finding missing angles:

• If you know two sides, use inverse trigonometric functions:
  • $\theta ={\sin }^{-1}\left(\frac{\text{opposite}}{\text{hypotenuse}}\right)$
  • $\theta ={\cos }^{-1}\left(\frac{\text{adjacent}}{\text{hypotenuse}}\right)$
  • $\theta ={\tan }^{-1}\left(\frac{\text{opposite}}{\text{adjacent}}\right)$

Remember these helpful mnemonics:

• SOH: Sine = Opposite/Hypotenuse
• CAH: Cosine = Adjacent/Hypotenuse
• TOA: Tangent = Opposite/Adjacent

Step 4: Solve Real-World Problems

Many SAT problems involve real-world scenarios like:

• Finding heights of objects using shadows or angles of elevation
• Calculating distances that cannot be measured directly
• Determining areas and volumes that involve right triangles

The key is to draw and label a diagram, identify the right triangle(s), and apply the appropriate formulas.

Step 5: Check Your Answer

Always verify that your answer makes sense in the context of the problem:

• Is the hypotenuse the longest side?
• Do the angles in your triangle sum to 180°?
• Are the units correct?
• For SAT problems, check if your answer matches one of the given options.

Part A: The Pythagorean Theorem

1. Find the length of the hypotenuse in a right triangle with legs of length 5 and 12.

2. If the hypotenuse of a right triangle is 13 units and one leg is 5 units, find the length of the other leg.

3. Determine whether a triangle with sides of length 9, 40, and 41 is a right triangle.

Part B: Special Right Triangles

4. In a 45°-45°-90° triangle, if each leg has length 7, find the length of the hypotenuse.

5. In a 30°-60°-90° triangle, if the shortest leg has length 8, find the length of the hypotenuse and the other leg.

Part C: Trigonometric Ratios

6. In a right triangle, if one angle is 38° and the hypotenuse is 20 cm, find the length of the side opposite to the 38° angle.

7. In a right triangle, if one angle is 45° and the adjacent side is 12 units, find the length of the opposite side.

8. Find the measure of angle θ in a right triangle if sin(θ) = 0.6.

Part D: Applications

9. A ladder 10 feet long leans against a vertical wall. If the bottom of the ladder is 6 feet from the base of the wall, how high up the wall does the ladder reach?

10. From a point on level ground 50 meters from the base of a tower, the angle of elevation to the top of the tower is 32°. Find the height of the tower.

Answers:

1. 13 units
2. 12 units
3. Yes $(9^2+40^2=41^2)$
4. $7\sqrt{2}$ units
5. Hypotenuse: 16 units; Other leg: $8\sqrt{3}$ units
6. 12.3 cm
7. 12 units
8. θ ≈ 36.9°
9. 8 feet
10. 31.4 meters

1. Confusing the Pythagorean Theorem: Some students incorrectly apply $a+b=c$ instead of $a^2+b^2=c^2$. Remember that it's the squares of the sides that have the relationship, not the sides themselves.

2. Misidentifying the Hypotenuse: The hypotenuse is always the side opposite to the right angle and is always the longest side in a right triangle. Students sometimes confuse the hypotenuse with one of the legs.

3. Mixing Up Trigonometric Ratios: Remembering which ratio is which can be challenging. Use the SOH-CAH-TOA mnemonic to keep them straight:

   • SOH: Sine = Opposite/Hypotenuse
   • CAH: Cosine = Adjacent/Hypotenuse
   • TOA: Tangent = Opposite/Adjacent

4. Forgetting to Identify the Reference Angle: When using trigonometric ratios, the terms "opposite" and "adjacent" are relative to the angle you're working with. Always identify which angle you're referencing before determining which sides are opposite and adjacent.

5. Incorrect Units: When solving word problems, students sometimes forget to include units in their final answer or mix up different units within the same problem.

6. Misapplying Special Triangle Ratios: For 30°-60°-90° triangles, students sometimes mix up which side is $x$, which is $x\sqrt{3}$, and which is $2x$. Remember that the shortest side (opposite to the 30° angle) is $x$, the middle side (opposite to the 60° angle) is $x\sqrt{3}$, and the hypotenuse is $2x$.

7. Assuming All Triangles Are Right Triangles: Not every triangle is a right triangle. Before applying the Pythagorean theorem or trigonometric ratios, verify that you're working with a right triangle.

8. Forgetting the Inverse Trigonometric Functions: When finding an angle given the sides, you need to use inverse trigonometric functions (${\sin }^{-1}$, ${\cos }^{-1}$, or ${\tan }^{-1}$). These are sometimes written as arcsin, arccos, and arctan.

9. Calculator Mode Errors: When using a calculator, make sure it's in the correct mode (degrees or radians) for your problem. Most SAT problems use degrees.