PLM MSTM 111 PDF

**PLM** **MSTM 111** **-[It] deals with the relations connecting the sides ang angles of a spherical triangle, is termed?** **Spherical trigonometry** **. It is the study of curved triangles, triangles drawn on the surface of sphere?. Spherical trigonometry** **In general, there are two spheres...

**PLM** **MSTM 111** **-[It] deals with the relations connecting the sides ang angles of a spherical triangle, is termed?** **Spherical trigonometry** **. It is the study of curved triangles, triangles drawn on the surface of sphere?. Spherical trigonometry** **In general, there are two spheres wherein spherical trigonometry is of great importance. What refers to connecting various points at surface of the earth, is known as?. Terrestrial sphere** **-In general, there are two spheres wherein spherical trigonometry is of great importance. What refers to connecting various *heavenly bodies lies like Sun,Moon,Planets, and Stars, is called* Celestial sphere** **- It is formed when a plane intersects a sphere in any direction Circle** **-. Itis formed when the plane of intersection passes through the center of the sphere? Great Circle** **-These are formed by the intersection of the sphere with a plane that does not pass through the center of the sphere? small circle** **- [Lines of latitude other than the Equator are examples of small circles](https://pages.uoregon.edu/jschombe/ast121/lectures/lec03.html) called as? small circle** **-** **An arc of 1' (one minute) is equal to one nautical [mile] distance on the surface of the terrestrial sphere (earth surface), which is approximately equal [to ?] 6,080 feet** **-Arc NS in fig.8.2 is equal to 180°** ; **how many in** **\_\_\_\_\_\_\_\_ nautical miles?. 10,800** **-In a Right Spherical Triangle ABC, C=90°, angle B=35°30', and side b=29°15'. Solve for remaining parts.111°05'** **- Solve for the missing parts of the Right spherical Triangle, Using rule II of Napier's Wheel,** **where:** **from rule II, Sin co-A=(cos a ) (cos (co-B), and Fig. 13: solve for the Angle A: 104°28'** **-From rule I: sin b =(tan co-A)(tan a) or, Refer to fig.13, Solve for the missing part-side b : 137°05'** **From Rule I: sin (co-B)= (tan(co-c))(tan a);Solve for te missing part, side c,** **74°58'** **- By** **Using Logarithmic or Useful table:** **-from Rule I and Fig.13, Sin co-c= (cos a)(cosb), solve for the missing part, side 83°13'** **From Rule 2: sin co-A = (tan co-c) ( tan b) and Fig. 13. Solve for the missing parts, angle A: 77°21'** **\-- From using rule I: sin co-B= (cos c0-A)(cos b), and Fig. 13, solve for the remaining part, angle B 62°15'** **-What does the word *Trigonometry means (from Greek) history?. trigonon \"triangle\" + metron \"measure\")*** **-.In Plane trigonometry, what type of triangle is of most interest?. the [right-angled triangle](https://www.mathsisfun.com/right_angle_triangle.html)** **It is defined as the ratio of the side opposite the angle to the hypotenuse. Sine function (sin) - It is defined as the ratio of the adjacent leg (the side of the triangle joining the angle to the right angle) to the hypotenuse Cosine function (cos),** **Refer to figure shown Use The Law of Cosines (angle version) to find angle C .... 57.9°** **Express the equation of the unit circle in its trigonometric fucntions with angle θ. cos²ϴ + sin²ϴ = 1** **-. Express the quotient relation of the tangent function in terms of angle θ ?.tanϴ = sinϴ / cos ϴ;** **- Express the Pythagorean identity and dividing both sides of the equation by sin²θ? cot²ϴ + 1 = csc²ϴ** **- Express the Pythagorean identity and by dividing both sides of the equation by cos²θ? 1 + tan²ϴ = sec²ϴ** **- EXPRESS the identity \[ 1 + cotθ \] / tan²θ in terms of sinθ and or cos θ? ( sin θ cos ²θ + cos³θ ) / sin³θ** \-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-- **\--A \_\_\_\_\_ *is a figure drawn on flat surface* bounded by a *straight or curve lines?* Plane Figure** **The distance around the plane figure is called Perimeter** **-The amount of space *measured in* cubic units, that an object or substance occupies is called?** **Volume** **-It [ ] is a closed plane figure bounded by line segments or broken lines,is known as? Polygon** **-The least number of broken lines that can form a Closed figure [ ] or Polygon is \_\_\_ three (3).** **A polygon with all the three sides and all the angles are equal is called a.Regular polygon** **-** **When two sides of a right triangle are given, to find the third side , what do you use? *Pythagorean Theorem.*** **-What i*s a three-dimensional figure bounded by surfaces, called?.* Geometric Solids** **-What is a polyhedron which has two congruent and parallel faces called bases, and whose other faces (called lateral faces) are parallelograms formed by joining corresponding vertices of the bases\-- Prism** **\--\> Find side *c.---* 13.4 cm** **fig.2. solve area of the triangle ABC by using Hero's (Heron's) Formula. A = √s(s-a)(s-b)(s-c)** 85. **q. cm** **-Fig 3..** **Find the volume of a rectangular prism 20 inches high, whose base is 13-inch square.** **Find : [The area of the base,(bxb), B=\_\_\_\_\_] 169 in²-** **-A container having a rectangular cr0ss-section of 8 ft. by 5 ft. has an altitude of 6 ft. Find the storage space of the container.also find the total external area of the container.** **The volume is ,V = l x w x h, 240ft³** **Total Area=2\[lw+lh+wh\]; 236ft²** **-** ***Find the volume and total area in ft³ and in.³respectively, of a cube of edge 2 ft 3 inches.see.fig.4*** **-The *volume of the cube is given by* e³: 11.39ft³** **- The total area, square inches, is 6e² 162in².** **What is a *branch* of *mathematics* that deals with the properties ,behaviors and solutions of points, lines,curves, angles, surfaces, and solids by means of *algebraic methods* in relation to *coordinate systems?*Analytic Geometry** **-refers to figures which deals with points on space.Solid analytic geometry** **What is considered as the simplest geometric figure** **i.e. it is the *shortest distance between two points.?*** **Straight Line** **-What is defined as a series of points P(x,y) in the Cartesian Coordinate Systems?Line** **-What is a formula that describes a series of any value of x that can be assigned in the equation so *that there is a corresponding value of y* ( for any value of y there is a corresponding value of x? The Equation of a line** **-The general equation of a line** is written in the **form ? Ax + By + C = 0,** **-What is the standard form of a linear equation which can be expressed as\_\_ where A and B cannot be both equal to zero. Ax + By = C,** **-(.SLOPE POINT FORM EQUATION OF A STRAIGHT LINE) *Write the equation of the line through the point* P( 3,-2)*, with slope* m = ¾ ; (y - y~1~ = m ( x - x~1~ ) 3x -4y -17 = o** **( TWO-POINT FORM EQUATION OF STRAIGHT LINE )Write the equation of the line as determined by the given points.** **SLOPE INTERCEPT FORM EQUATION OF STRAIGHT LINE) ( 5 points). Find the equation of the line as determined by the slope *m* and *y-intercept, b*. 3x -4y +20 = 0** **- *INTERCEPT FORM EQUATION OF STRAIGHT LINE: Determine the equation of the line with the given intercepts:*** **- {EQUATION OF A CIRCLE\]; Center at (h,k)\] Determine the standard equation of a circle with center at (3,-2) and with radius equal to 3 units,WHERE: (x-h)² + (y-k)² = r²( x-3)² + ( y + 2 )² = 9** **-Studying that involves understanding the geometry of triangles on the surface of a sphere where none of the angles are right angles, is termed ?. Oblique Spherical Triangles** **- For a spherical triangle with sides (a), (b), and (c), and angles (A), (B), and (C) opposite these sides,** - **the Spherical Law of Cosines (for side), what is the derived formula for side c cos(c)=cos(a)cos(b)+sin(a)sin(b)cos(C)** **For a spherical triangle with sides (a), (b), and (c), and angles (A), (B), and (C) opposite these sides,** - **the Spherical Law of Cosines (for side), what is the derived formula for side b only.?** **-For a spherical triangle with sides (a), (b), and (c), and angles (A), (B), and (C) opposite these sides,** - **the Spherical Law of Cosines (for side), what is the derived formula for side a only?. cos a = cos b cos c + sin b sin c cos A** **For a spherical triangle with sides (a), (b), and (c), and angles (A), (B), and (C) opposite these sides,** **the Spherical Law of Cosines (for side), what is the derived formula for angle C only?cos C=-cos A cos B + sin A sin B cos c** **-For a spherical triangle with sides (a), (b), and (c), and angles (A), (B), and (C) opposite these sides,** **the Spherical Law of Cosines (for side), what is the derived formula for angle A only?** ***cos A= - cos B cos C + sin B sin C cos a*** ***\-\--\>*For a spherical triangle with sides (a), (b), and (c), and angles (A), (B), and (C) opposite these sides,** **the Spherical Law of Cosines (for side), what is the derived formula for angle B only?cos B=-cos A cos C + sin A sin C cos b** **-To derive the formula for sine Law, in a Oblique spherical triangle[,] we consider any spherical triangle with vertices at ABC and sides labeled as a, b, and c. \[fig.10.1\];** **Draw an arc of the great circle from vertex B perpendicular to the opposite side b at point D as shown in [fig. 10.1]** **Write the derived Formula for Sine Law? "sin a / sin A"= "sin b/sin B" = "sin c / sin C"** **[Problem for Case V and case VI (SSA) and (AAS)-]** **Find the opposite side of angle A of a spherical triangle with A=35° 18'. B=122° 45'; and *b= 78° 25'*** - **Reqd: missing side a:** **Using sine Law:** **-the missing side a, 42° 18'...** **-Fig. 12.[SOLVE PROBLEM CASE 111 AND CASE IV:]** - **Using cosine law for sides, Find [side c] of a spherical triangle.** - **cos c= cos a cos b +sin a sin b cos C** **62° 31'** **Find the angle A of a spherical triangle ABC-[Case 1 ,Case 2 (SSS ) and (AAA-)]** - **cos A = \[cos a- cos b cos c\] / sin b sin c 56° 06'** **-Right spherical Traingle ABC , [CASE 111 AND CASE IV:]** - **cos b= cos a cos c +sin a sin c cos B 59.78'..** **-Find the angle B of a spherical triangle ABC-[Case 1 ,Case 2 (SSS ) and (AAA-)]** - **cos B = \[cos b- cos a cos c\] / sin a sin c 41 ° 39'** **[Case V nd case VI (SSA) and (AAS)]** - **sin a/sin A = sin b / sin B 42° 18'...**