“Right Triangle” Trigonometry

Exercises

1. For each of the following right triangles, determine the lengths of the unlabelled sides and measures of the unlabelled angles.

   

2. For each of the following triples of numbers, determine if there exists a triangle with those numbers as the lengths of its sides, and if so, determine if that triangle will be acute, right, or obtuse.

   \(\bigl\{6,7,8\bigr\}\)

   \(\bigl\{6,8,10\bigr\}\)

   \(\bigl\{6,8,11\bigr\}\)

   \(\bigl\{2,3,4\bigr\}\)

   \(\bigl\{2,3,6\bigr\}\)

   \(\bigl\{1,2,3\bigr\}\)

   \(\bigl\{5,12,12\bigr\}\)

   \(\bigl\{5,12,14\bigr\}\)

   \(\bigl\{5,12,17\bigr\}\)

3. Harriet the hiker is hiking a hill that has an incline (grade) of 17°. If Harriet has climbed a distance of 2300′ along the incline, how much altitude (in feet) has she gained?
4. Larry the lineman has just secured a 50′ guy-wire from the top of an electrical pole to the level ground away from the pole’s base. If the wire makes a 56° angle with the ground, how far from the pole is the wire anchored?

Problems & Challenges

1. Suppose a right triangle has a pair of side-lengths measuring 7″ and 9″. What are the two possibilities for the measure of the third side?
2. Consider three circles having radii measuring one, two, and three, that are each mutually tangent to each other. What is the area of the sector of the smallest circle that is cut off by the two segments joining its center to the centers of the others?
3. Mr Goat is tethered to a stake in the ground at the concave corner of an “L”-shaped shed. The ground outside the shed is covered with lush delicious ground-cover — grasses and clovers and creeping red thyme — that Mr Goat would like to eat. Considering the length of Mr Goat’s tether and the dimensions of the shed indicated in the image, what is the total area of ground cover that Mr Goat can raze? Challenge: Suppose instead of that inner corner, Mr Goat were tethered to the upper-right corner indicated with a small square. Now what is the total area of ground cover that Mr Goat can raze?
4. Suppose a right triangle of area \(5\) contains an acute angle \(\theta\) such that \(\tan(\theta) = 2.\) What are the side-lengths of the triangle?
5. TK some of these require the inverse trig functions ... which are next class's topic.
6. Angeline is standing on level ground, some distance away from the base of St Mary’s Medical Center, which, at 150′ tall, is the tallest building in Grand Junction. Looking up at the very top of the building, Angeline pulls out her trusty inclinometer and measures that the angle of elevation of her line of sight is 72°. How far away is Angeline from the base of St Mary’s?
7. Dr Baxter, bored with his daily commute home from work, decides to build a zip line from the roof of St Mary’s Medical Center to the nearby park that is in the direction of his house. St Mary’s is about 150′ tall. He decides he can afford to set up a zip line with a braking system, in which case experts recommend a 6% grade as the steepest direction of decline that is still safe. Sketching out the design, Dr Baxter has to figure out two things: assuming the zip line will be fairly taught, what angle will it make with the level ground in the park? And how long does the line itself need to be?
8. Celina has climbed a tree! The tree is 33′ tall, growing straight out of level ground. She spots her friend away from the base of the tree and, pulling out her trusty inclinometer, measure the angle of depression towards her friend to be 53°. If her fiend pulls out an inclinometer too and measures the angle of inclination towards Celina, what should the measurement be? How far away from the base of the tree is Celina’s friend?
9. The Ladder Association advises that, for safety’s sake, when setting up a ladder to be climbed, the base of ladder should make at least a 75° angle with the ground. Demetrius needs to buy a ladder to install rain gutters on his home. He’s about 6′ tall, and the roofline of his house is 14′ off the ground.
   1. If Demetrius is dedicated to always setting up his ladder safely, what’s the shortest possible ladder he should buy?
   2. While out shopping for ladders, Demetrius notices there are painters working on ladders outside the hardware store. He notices that one painter’s ladder is 11′ long and positioned with its feet 3′ from the base of the wall. Should Demetrius go talk to the painter about safety?
   3. Finally looking at ladders in the ladder aisle of the hardware store, Demetrius notices that there are warnings on each ladder to not stand on top, or stand on the top rung. On every ladder for sale, the highest rung that’s safe to stand on is 2′ from the top of the ladder. What’s the shortest possible ladder that Demetrius, still dedicated to safety, should buy?
10. Edith is flying a kite. She’s let out 123′ of kite string and, using her trusty inclinometer, measures that the kite’s angle of elevation from her position is 36°. Assuming the kite string is taught, what’s the kite’s altitude?
11. Finneas is hiking to the top of a tall hill that has a constant incline of 25°. After reaching the top of the hill, his altimeter tells him his altitude changed by 6000′ since starting his hike. How long was the hike?
12. Garamond is standing sad and alone in an empty parking lot. He is 5′10″ tall, and his shadow is 11′ long from his feet to the tip of his shadow-head. What is the angle of inclination of the sun?
13. Henrietta wants to set up a camera to automatically record an aerial fireworks show. She positions the camera on the level ground about 500′ from where the fireworks will be launched, and Henrietta knows that, on average, these fireworks will launch 200′ in the air. At what angle of inclination should she tilt the camera to ensure she gets a good recording of the show?