5. Practice Exercises

A tree casts a shadow of 10 m when the angle of elevation of the sun is 45°. Find the height of the tree.

To find the height of the tree, we can use the concept of trigonometric ratios. Since we know the angle of elevation is 45 degrees and the length of the shadow is 10 meters, we can use the tangent function. The tangent of an angle in a right triangle is the ratio of the opposite side (height of the tree) to the adjacent side (length of the shadow). So, tan(45°) = height / 10 m. Since tan(45°) is equal to 1, we can set up the equation: 1 = height / 10. Solving for height gives us a height of 10 meters.

Imagine you're trying to figure out how tall a tree is just by looking at its shadow on a sunny day. If the sun is positioned so that your shadow on the ground is as long as the tree's, it makes sense that the tree would be just as tall as the shadow. So, standing by the tree, you can use simple geometry to find its height.

In a triangle, one angle is 90°, and the two other sides are 8 cm and 15 cm. Find the hypotenuse.

This exercise requires the use of the Pythagorean Theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Here, we have the lengths of two sides: 8 cm and 15 cm. So, using the formula, we can find the hypotenuse (let's call it c): c² = 8² + 15². This calculates to c² = 64 + 225 = 289. Taking the square root of both sides gives us c = √289, which equals 17 cm. Therefore, the hypotenuse is 17 cm long.

Think about a ramp. If one side is 8 m high and the bottom of the ramp is 15 m away from the wall, the ramp itself acts as the hypotenuse. You can figure out how long the ramp needs to be to reach the top by applying this theorem, which can be crucial when constructing wheelchair ramps or escalators.

Prove the Basic Proportionality Theorem using a diagram.

The Basic Proportionality Theorem, or Thales' Theorem, states that if a line is drawn parallel to one side of a triangle, intersecting the other two sides, it divides those sides proportionally. To prove this, you can create a triangle and draw a line parallel to one side. Then, label the intersection points. You can use similar triangles to establish that the ratios of the segments created by the parallel line are equal. If AB and CD are the intersecting line segments, the ratio of AC/AD will equal the ratio of BC/BD. This can be demonstrated through the triangle similarity rules.

Picture a street with traffic lights set up along a road that makes an angle with a crossroad. The traffic light pole creates a triangle between itself and where it touches the ground. If you could draw a straight road parallel to another and analyze how the distances between poles relate to each other, this relationship exemplifies the theorem in a practical scenario.

Show that in an isosceles triangle with equal sides of 5 cm and base 6 cm, the base angles are equal.

In an isosceles triangle, the two sides that are equal (in this case, both 5 cm) will have angles that are also equal. To see this visually, draw the triangle and label the angles opposite the equal sides as A and B. You can use the Triangle Sum Theorem here, which states the sum of the angles in a triangle is 180 degrees. By finding the third angle (angle C), you can show that A = B by calculating: A + B + C = 180°, where C can be calculated knowing C is opposite the unequal side of 6 cm.

Think about a pair of speakers that are identical in size. If placed in a room making the same angle towards a listener, it guarantees that the sound wave from both speakers reaches the listener at the same time, creating an equal sound experience—a concept paralleling how angles in geometry function.

If sin θ = 0.6, find cos θ and tan θ using identity.

To find cos θ and tan θ using the given sin θ = 0.6, we can utilize the Pythagorean identity, which states that sin²θ + cos²θ = 1. First, we can find cos θ: 0.6² + cos²θ = 1 leads to 0.36 + cos²θ = 1. Thus, cos²θ = 1 - 0.36 = 0.64, and taking the square root gives us cos θ = ±0.8. Now, to find tan θ, we use the formula tan θ = sin θ / cos θ. Hence, tan θ = 0.6 / 0.8 = 0.75.

Imagine you are hiking up a hill. The height you gain compared to the distance you walk forms a right angle triangle. Utilizing your height (sin θ) against the diagonal path you take represents these trigonometric functions, helping calculate how steep or flat your hike truly is—making planning much easier.

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