guÍa de aprendizaje no 12 - asdrumath...2020/10/10 · swokowski, w. (2011). algebra y...

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GUÍA DE APRENDIZAJE No 12 ÁREA DE MATEMÁTICAS – GRADO DÉCIMO Colegio

Nombre del Estudiante: Curso

10°

DD MM

AA

2020

Tema: Solving oblique triangles

TIME: 5 Units

RESOURCES: Notebook and Math book

OBJECTIVE: To understand the law of sine and cosine, to demonstrate proficiency in solving triangles using sine and cosine law, to apply the sine and cosine laws to solve trigonometric problems.

AUTONOMY INDICATOR: To learn how to generalize about the learning processes of the different subjects: Characterizes the different subjects and the ways to proceed within each one of them.

LEARNING STRATEGY: Teacher’s explanation, reading tables and graphs.

1. CALENTANDO MOTORES (WARMING UP THE ENGINE)

The first Colombian satellite “Freedom One”, launched from Kazajistan in order to take photos and to send compressed images and temperature data signals, was orbited in year 2007. The satellite´s shape is a cube that weights 1 Kg and its edge measures 10 cm.

If the satellite is observed simultaneously from the observation station in Bogota (elevation angle 68°) and from the observation station in Cucuta (elevation angle 54°). How far is the satellite from Bogota? How far is the satellite from Cucuta?

Make a chart of the situation and determine the possibility of solving the problem by applying trigonometric ratios:

2. LO QUE SÉ (WHAT I KNOW)

a. Determine the perimeter and area of the quadrilateral ABCD, if AB and DC are parallel tangents to the circle.

𝑨𝑩̅̅ ̅̅ = 𝟗 𝑶𝑨̅̅ ̅̅ = 𝟔, 𝟓

𝑪𝑫̅̅ ̅̅ = 𝟒

b. Solve each of the following triangles.

I. EXPLORANDO (EXPLORING TIME)

1 UNIT

Aprobado por: COORDINADOR DE DEPARTAMENTO PEDAGÓGICO V1 de 01/09/2020 Página 2 de 7

3. CAJA DE HERRAMIENTAS (TOOLBOX)

a. LAW OF SINE

For each of the given triangles develop each of the following exercises in the notebook:

A. Trace the height of the triangle with respect to the segment AB.

B. Write the Seno ratio for each one of the triangles formed.

C. Calculate the height value in each of the ratios given.

D. Set the equations equal and calculate the unknown value.

Resolution of triangles according to the criteria: Angle,

Angle, Side (AAS)

Resolution of triangles according to the criteria Side, Side,

Angle (SSA)

b. LAW OF COSINE

For each of the given triangles develop each one of the following exercises in the notebook:

A. Draw the height of the triangle with respect to the segment AB.

B. Use the given angle to determine the value of the adjacent side in the formed right triangle.

C. Determine the value of the side b projection in terms of the data given and obtained.

D. Apply the Pythagoras Theorem in both triangles.

E. Calculate h2 value in each one of the equations and then set them equal.

F. Develop, simplify and calculate the unknown value.

Resolution of triangles according to the criteria: Side,

Angle, Side (SAS)

Resolution of triangles according to the criteria Side, Side,

Side (SSS)


c. MI DESAFIO ES… (MY CHALLENGE IS…)

Bearing in mind the previous information, the one offered by the teacher and your previous knowledge, write your learning goal in the notebook. ___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

1. Read the following information.

Law of Cosine

The law of cosine is a formula that relates the three sides of a triangle to the cosines of a given angle. This formula allows you ✓ To calculate the side length of non-right triangle as long

as you know two sides and an angle. ✓ To calculate any angle of a triangle if you know all three

side lengths. The formula for the law of cosines expresses the following relationships between the sides, angles of any triangle:

βCoscbcba 2-+= 222

θCoscacab 2-+= 222

αCosbabac 2-+= 222

Law of Sine

The law of sine provides a formula that relates the sides with the angles of a triangle. This formula allows you to (in a relatively easy way), find the side length or the angle of any triangle.

c

Sin

b

Sin

a

Sin θβα==

θβα Sin

c

Sin

b

Sin

a==

2. Based on the reading, solve the practice problems.

Practice Problem one: Use the law of cosines formula to calculate the length of side A.

Practice Problem one: Use the formula for the law of sine to determine the measure of angle b below.

II. MANOS A LA OBRA (HANDS ON)

1 UNIT

Step 1: set up the formula:

Step 2: calculate:

Step 2: calculate:



Practice Problem two: Use the law of cosines to calculate x.

Practice Problem two: Use the formula for the law of sine to determine the unknown side length.

1. Find the value of “𝑥” in all of the following triangles.

2. Using the figure,

2.1 Find the missing values of the angles and sides of the triangle.

a. °=== 31C1,2b;6a

b. °=== 115A7,11c;3,12b

c. °=== 1,112B8,7c;3,5a

d. 59c173b;192a ===

e. 7,6c4,9b;8,5a ===

2.2 Find the area of the following triangles:

a. °=°== 37B65A;105a

b. °=°== 52A26B;89b

c. °=°== 2,68C3,54A;8,35a

d. °=°== 6,104C4,43B;6,72c

e. °=°== 9,71C2,43A;9,17b

III. CONSTRUYAMOS JUNTOS (LET´S CONSTRUCT TOGETHER)

1 UNIT

Step 2: calculate: Step 2: calculate:



http://www.mathwarehouse.com/trigonometry/law-of-sines/formula-and-practice-problems.php


3. Find the measure of

the angles of the following triangle.

4. The sides of a triangle measure 4 cm, 5cm and 7cm. What is it area? (Hint: use the law of cosine to find one angles and use it to find the height of the triangle)

5. A boat travels 40 km between the cities A and B, heading north-west 65°. From the city B it is directed to another city C at 30° North-east 250km distant, as shown in the figure. Calculate the distance between cities A and C, and the direction to be taken to return the boat if it goes directly between the cities.

6. A tower is inclined 10° vertical and held by a cable from a point P 15 meters from the base of the tower. If the elevation of the wire angle is 25°. Calculates the wire length and the height of the tower.

7. A person observes an airplane and a boat from the top of a lighthouse, as shown in the figure. What is the distance from the boat to the viewer, and the boat to the aircraft?


8. A tree is observed from two opposite points separated 250 meters with elevation angles of 30 ° and 25 °. What is the height of the tree and how far is the height of the tree and the distance of each point of observation to the top?

9. The diagonals of a parallelogram are 10 cm and 12 cm, and the angle is 48 ° 15 '. Find the sides of the parallelogram.

10. Find the measure of the angle A.

In order to prepare for the evaluation, answer the following questions in the most responsible way:

PERFORMANCE YES NO WHY HOW CAN I IMPROVE?

I can solve right triangles _____________________ _____________________ _____________________

____________________________ ____________________________ ____________________________

I understand the law of sine and

cosine.

_____________________ _____________________ _____________________

____________________________ ____________________________ ____________________________

I apply the laws of the sine and cosine in solving problems.

_____________________ _____________________ _____________________

____________________________ ____________________________ ____________________________

I can demonstrate ability in solving triangles applying the law of sine and

cosine.

_____________________ _____________________ _____________________

____________________________ ____________________________ ____________________________

IV. PROGRESANDO EN EL DESAFIO (PROGRESSING IN THE CHALLENGE)

1 UNIT


Creatively demonstrate Heron's formula on a sheet of paper. Add two examples.

Solve the following problems:

1. When the Moon rotates around the Earth, the side facing the Earth is usually only partially illuminated by the Sun. The phases of the Moon describe how much of the surface appears to be in sunlight. An astronomical measure is given by the fraction 𝐹 of the lunar disk that is illuminated. When the angle between the Sun, Earth and Moon is 𝜃 (10 ≤ 𝜃 ≤ 360°), then

𝐹 =1

2(1 − 𝐶𝑜𝑠 𝜃)

Determine the angles 𝜃 corresponding to the following phases:

a. 𝐹 = 0 (new moon) b. 𝐹 = 0, 25 (growing quarter) c. 𝐹 = 0, 5 (first or last quarter) d. 𝐹 = 1 (full moon)

2. If a projectile is fired at velocity 𝑣0 at an angle 𝜃, then its range, the horizontal distance it travels (in feet), is modeled by the

function:

𝑅(𝜃) =𝑣0

2 𝑆𝑒𝑛 2𝜃

32

If the initial velocity is 𝑣0 = 2200 𝑓𝑡

𝑠⁄ , what angle (in degrees) should be chosen for the projectile to hit the target on the ground 500 000 feet away? REFERENCES:

Buitrago, et al (2014), Matemáticas 10 Santillana-compartir. Bogotá: Santillana. http://www.mathwarehouse.com/trigonometry/law-of-sines/ambiguous-case-of-law-of-sines.php Stewart, J. Redlin, L. & Watson, S. (2012). Precálculo: Matemáticas Para el Cálculo. 6ta edición. Swokowski, W. (2011). Algebra y Trigonometría con Geometría Analítica 13ª edición.

Administradores de Programa: Equipo Pedagógico 10°

Control de cambios

Versión Descripción del cambio Elaborado por Aprobado por

V2 02/09/2019

Se ajustan títulos y diagramación de acuerdo con nuevas directrices.

Docente Administrador Coordinador de área

V. RETOMANDO ANDO (CHECK POINT)

1 UNIT
http://www.mathwarehouse.com/trigonometry/law-of-sines/ambiguous-case-of-law-of-sines.php

guÍa de aprendizaje no 12 - asdrumath...2020/10/10 · swokowski, w. (2011). algebra y...

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