We need to find cos(θ) when the adjacent side is 8 and the opposite side is 6. First, we must find the hypotenuse using the Pythagorean theorem: h² = 8² + 6² = 64 + 36 = 100, so h = 10. Using SOH-CAH-TOA, cos(θ) = adjacent/hypotenuse = 8/10 = 4/5. A frequent mistake is using the wrong sides in the ratio or forgetting to calculate the hypotenuse first. Always identify all three sides of the triangle before applying any trigonometric ratio.

We need to find the height of a building when it casts a 30-meter shadow and the sun's angle of elevation is 60°. The height is opposite to the 60° angle, and the shadow is adjacent to this angle. Using SOH-CAH-TOA: tan(60°) = height/30, so height = 30 × tan(60°) = 30 × √3 ≈ 30 × 1.732 = 51.96 meters. A common mistake is using sine or cosine instead of tangent, or confusing which sides are opposite and adjacent to the angle of elevation. In shadow problems, the tangent ratio connects the height and shadow length through the elevation angle.

We need to find the height of the kite above the ground when it's at a 45° angle of elevation with a 20-meter string. The height is the side opposite to the 45° angle, and the string is the hypotenuse. Using SOH-CAH-TOA: sin(45°) = height/20, so height = 20 × sin(45°) = 20 × (√2/2) = 20/√2 = 10√2 ≈ 14.14 meters. A common error is using cosine instead of sine or confusing the angle of elevation with another angle. In elevation problems, the height is always opposite to the angle of elevation.

We need to find the hypotenuse when given the adjacent side to angle B and the angle itself. Since we have the adjacent side and need the hypotenuse, we use the cosine ratio: cos(B) = adjacent/hypotenuse. Setting up the equation: cos(45°) = 7/hypotenuse, so hypotenuse = 7/cos(45°) = 7/(√2/2) = 7 × √2 = 9.9 units approximately. A common mistake is using sine instead of cosine or confusing which side is adjacent to the given angle. Remember that in a 45° right triangle, the adjacent and opposite sides are equal, and the hypotenuse is √2 times either leg.

We need to find the length of side DF in a right triangle where ∠D = 90°, ∠E = 45°, and DE = 5 units. Since ∠D is the right angle, sides DE and DF are the legs, and EF is the hypotenuse. In a 45-45-90 triangle, the two legs are equal in length. Since DE = 5 units and this is a 45° triangle, DF must also equal 5 units. A common error is assuming one of the legs is the hypotenuse or forgetting the special properties of 45-45-90 triangles. In isosceles right triangles, always remember that the two legs are congruent.

We need to find cos(θ) where θ is the angle at the origin in a right triangle with vertices at (0,0), (3,0), and (0,4). Using SOH-CAH-TOA, cosine is defined as cos(θ) = adjacent/hypotenuse. The adjacent side (horizontal leg) has length 3, the opposite side (vertical leg) has length 4, and the hypotenuse has length √(3² + 4²) = √(9 + 16) = √25 = 5. Therefore: cos(θ) = adjacent/hypotenuse = 3/5. A common error is confusing the adjacent and opposite sides relative to the angle at the origin. When working with coordinate geometry, carefully identify which sides correspond to adjacent and opposite for the specified angle.