Trigonometry means "triangle measurement." This branch of mathematics reveals the precise relationships Yahuah built into creation. From the angles of light to the heights of mountains, trigonometry helps us understand and measure His world with precision.
Yahuah is the master mathematician who designed all the precise relationships we discover in trigonometry.
A right triangle has one angle that measures exactly 90° (a right angle). This special triangle is the foundation of all trigonometry.
A right triangle with angle θ (theta) at the bottom left
Right triangles appear everywhere in creation and construction. The pyramids, buildings, and even the shadows cast by the sun form right triangles. Yahuah designed these consistent relationships into His creation.
Key Points:
The Pythagorean Theorem describes a relationship that Yahuah built into every right triangle in creation:
Where a and b are the legs, and c is the hypotenuse
This means: If you square the lengths of both legs and add them together, you get the square of the hypotenuse.
A right triangle has legs of 3 and 4. Find the hypotenuse.
Solution:
a² + b² = c²
3² + 4² = c²
9 + 16 = c²
25 = c²
c = √25 = 5
Some right triangles have all whole number sides. These are called Pythagorean triples:
Yahuah designed right triangles with consistent relationships. No matter how large or small a right triangle is, if the angles are the same, the ratios of the sides are the same!
A ratio is a comparison of two numbers. In trigonometry, we compare sides of triangles.
Sine = Opposite / Hypotenuse
Cosine = Adjacent / Hypotenuse
Tangent = Opposite / Adjacent
Similar triangles (same angles, different sizes) have the same ratios. This is why trigonometry works! A 30° angle has the same sine value whether the triangle is the size of your hand or the size of a pyramid.
Sine tells us the ratio of the side opposite an angle to the hypotenuse.
In a right triangle, the side opposite angle A is 3 and the hypotenuse is 5.
sin(A) = 3/5 = 0.6
| Angle | sin(θ) | Decimal |
|---|---|---|
| 0° | 0 | 0 |
| 30° | 1/2 | 0.5 |
| 45° | √2/2 | ≈0.707 |
| 60° | √3/2 | ≈0.866 |
| 90° | 1 | 1 |
Cosine tells us the ratio of the side adjacent (next to) an angle to the hypotenuse.
In a right triangle, the side adjacent to angle A is 4 and the hypotenuse is 5.
cos(A) = 4/5 = 0.8
| Angle | cos(θ) | Decimal |
|---|---|---|
| 0° | 1 | 1 |
| 30° | √3/2 | ≈0.866 |
| 45° | √2/2 | ≈0.707 |
| 60° | 1/2 | 0.5 |
| 90° | 0 | 0 |
Notice: cos(θ) = sin(90° - θ). These are complementary!
Also: tan(θ) = sin(θ) / cos(θ)
Tangent compares the opposite side to the adjacent side. It doesn't use the hypotenuse!
In a right triangle, the opposite side is 3 and the adjacent side is 4.
tan(θ) = 3/4 = 0.75
| Angle | tan(θ) | Decimal |
|---|---|---|
| 0° | 0 | 0 |
| 30° | √3/3 | ≈0.577 |
| 45° | 1 | 1 |
| 60° | √3 | ≈1.732 |
| 90° | undefined | ∞ |
When you know an angle and one side, you can find other sides using trigonometry!
Angle = 30°, Hypotenuse = 10. Find the opposite side.
Solution:
sin(30°) = opposite / 10
0.5 = opposite / 10
opposite = 0.5 × 10 = 5
Angle = 60°, Hypotenuse = 12. Find the adjacent side.
Solution:
cos(60°) = adjacent / 12
0.5 = adjacent / 12
adjacent = 0.5 × 12 = 6
When you know two sides but need to find an angle, use inverse trig functions:
θ = sin⁻¹(opposite/hypotenuse)
θ = cos⁻¹(adjacent/hypotenuse)
θ = tan⁻¹(opposite/adjacent)
Also written as arcsin, arccos, arctan
Opposite = 5, Hypotenuse = 10. Find angle θ.
Solution:
sin(θ) = 5/10 = 0.5
θ = sin⁻¹(0.5)
θ = 30°
Two special right triangles appear frequently in mathematics. Memorizing their ratios saves time!
An isosceles right triangle with two 45° angles.
Side Ratios: 1 : 1 : √2
Half of an equilateral triangle.
Side Ratios: 1 : √3 : 2
Trigonometry isn't just abstract math—it's a tool Yahuah gave us to measure and understand His creation!
Yahuah is the ultimate surveyor and mathematician!
You stand 50 feet from a tree. The angle to the top of the tree is 60°. How tall is the tree?
Solution:
tan(60°) = height / 50
1.732 = height / 50
height = 1.732 × 50 ≈ 86.6 feet
1. hypotenuse | 2. 90 | 3. legs | 4. 90° | 5. opposite
1. c = 10 (6² + 8² = 100, √100 = 10)
2. a = 12 (13² - 5² = 169 - 25 = 144, √144 = 12)
3. b = 12 (15² - 9² = 225 - 81 = 144, √144 = 12)
4. Yes (6² + 8² = 36 + 64 = 100 = 10²)
5. 6 feet (10² - 8² = 100 - 64 = 36, √36 = 6)
1. SOH-CAH-TOA
2. opposite, hypotenuse
3. adjacent, hypotenuse
4. opposite, adjacent
1. 5/13 ≈ 0.385 | 2. 8/10 = 0.8 | 3. 0.5 or 1/2 | 4. 1
1. 12/13 ≈ 0.923 | 2. 6/10 = 0.6 | 3. 0.5 or 1/2 | 4. 1
1. 5/12 ≈ 0.417 | 2. 8/8 = 1 | 3. 1
4. At 90°, the adjacent side would be 0, and division by 0 is undefined.
1. ≈ 7.07 (10 × sin 45° = 10 × 0.707)
2. ≈ 4.62 (8 × tan 30° = 8 × 0.577)
3. ≈ 10.4 (9 ÷ sin 60° = 9 ÷ 0.866)
1. 45° | 2. 60° | 3. 30°
1. 5√2 ≈ 7.07 | 2. 4√3 ≈ 6.93 | 3. 8
4. 45°, 45°, and 90°
1. ≈ 14° (tan⁻¹(3/12) = tan⁻¹(0.25))
2. ≈ 18.8 feet (20 × sin 70°)
3. ≈ 17,320 feet (10,000 ÷ tan 30°)