TRIGONOMENTRY

Ensure your calculator must be in degree mode. If you are not sure

press [ON] [MODE] [MODE] [MODE] [MODE] “converted to degree” 

(1). If sin A = ⁴/₅ and cos B = ⁵/₁₃ evaluate cos (A-B)

(A) ⁵⁶/₆₅ (B) ⁶³/₆₅ (C) ¹²/₁₃ (D) ¹³/₁₂

Solution:-Using CASIO fx-991MS calculator

[ON] [SHIFT] [sin^-1] [4] [a^b/_c] [5] [SHIFT] [STO] [A] (53.13010235)

[AC] [SHIFT] [cos^-1] [4] [a^b/_c] [5] [SHIFT] [STO] [B] (67.38013505)

Evaluate cos (A-B)

[AC] [cos] [(]  [ALPHA] [A] [-] [ALPHA] [B] [)] [=] (0.969230769) press key [ab/c] (⁶³/₆₅)

The correct option is Option “B”

(2). If cos x = ¹²/₁₃ 0 ≤ X ≤ 90⁰ find sin² x + (1 + cos x)²

(A) ²⁵/₁₃ (B) ⁵⁰/₁₃ (C) ⁵⁰/₁₆₉ (D) 2

(ii). Cosec x + sin x

(A) ⁶⁴/₆₅ (B) ²⁵/₁₃ (C) ²⁵/₁₆₉ (D) ¹⁹⁴/₆₅

(iii). Cot x + 2 sec x

(A) ¹³⁷/₃₀ (B) ³⁷/₃₀ (C) ²⁰⁹/₆₀ (D) ²⁹/₆₀

(iv) sec x + tan x – 1

(A) - ½ (B) 2 (C) ½ (D) 6

(V) Cosec x + cot x – 1

(A) 3 (B) 4 (C) 5 (D) 6

Solution:-Using CASIO fx-991MS calculator

[ON] [SHIFT] [cos^-1] [1] [2] [a^b/_c] [1] [3] [SHIFT] [STO] [X](x = 0.394791119)

(i). Sin² x + (1 + cos x)²

[AC] [(] [sin] [ALPHA] [X] [)] [x²] [+] [(] [1] [+] [cos] [ALPHA] [X] [)] [x²] [=] (3.846153846) press [a^b/_c] (3 ¹¹/₁₃) press [SHIFT] [^d/_c] (⁵⁰/₁₃)

The correct option is “B”

(ii). Cosec x + sin x

But cosec x = ¹/_{sin x}

[AC] [1] [a^b/_c] [sin] [ALPHA] [X] [+] [sin] [ALPHA] [X] [=] (2.984615385) press [a^b/_c] (2 ⁶⁴/₆₅) press [SHIFT] [^d/_c] (¹⁹⁴/₆₅)

The correct option is “D”

(iii). Cot x + 2 sec x

But cot x = ¹/_{tan x} , sec x = ¹/_{cos x}

[AC] [1] [a^b/_c] [cos] [ALPHA] [X] [+] [2] [(] [1] [a^b/_c] [cos] [ALPHA] [X] [)] [=] (4.566666667) press [a^b/_c] (4 ¹⁷/₂₀) press [SHIFT] [^d/_c] (¹³⁷/₃₀)

The correct option is “A”

(iv) Sec x + tan x – 1

[AC] [1] [a^b/_c] [cos] [ALPHA] [X] [+] [tan] [ALPHA] [X] [-] [1] [=] (0.5) press [a^b/_c] (½) 

The correct option is “C”

(v) Cosec x + cot x – 1

[AC] [1] [a^b/_c] [sin] [ALPHA] [X] [+] [(] [1] [a^b/_c] [tan] [ALPHA] [X] [-] [1] [=] (4)

The correct option is “D”

(2). Simplify ¹/_{1 – sin x} + ¹/_{1 + sin x} 

(A) 1 – sin² x (B) 2cos² x (C) 1 – cos² x (D) 2sec² x

Solution:-Using CASIO fx-991MS calculator

Make x = any number of your choice i.e. x = 10

ensure your calculator must be in degree mode. If you are not sure

press [ON] [MODE] [MODE] [MODE] [MODE] “converted to degree” 

[AC] [10] [SHIFT] [STO] [X] (X=10)

[AC] [1] [a^b/_c] [(] [1] [-] [sin] [ALPHA] [X] [)] [+] [1] [a^b/_c] [1] [+] [sin] [ALPHA] [X] [)] [=] (2.062182408)

If any of the option gives 2.062182408 it is correct but if it didn’t give 2.062182408 it is wrong

Checking option (A) 1 – sin² x

But sin² x = (sin x)² 

[AC] [1] [-] [(] [sin] [ALPHA] [X] [)] [x²] [=] (0.96984631)

Option “A” is wrong

Checking option (B) 2cosec² x

But cosec² x = ¹/_(sin² _x) = ¹/_(sin² _x) 

[AC] [1] [a^b/_c] [(] [sin] [ALPHA] [X] [)] [x²] [=] (66.32687496)

Checking option (C) 1 – cosec² x

[AC] [1] [a^b/_c] [(] [sin] [ALPHA] [X] [)] [x²] [=] (-32.16343748)

Checking option (D) 2sec² x  But sec² x = ¹/_{cos x} 

[AC] [2] [(] [1] [a^b/_c] [cos] [ALPHA] [X] [)] [x²] [=] (2.62182408) 

The correct option is “D”

(4). Given that x is an acute angle such that sin x = ^y/_m  cos x is

(A) y² – m²  (B) √ (y² – m² (C) √m² – y² (D) None

            m                    m                  m

Solution:-Using CASIO fx-991MS calculator

Make m and y = any number of your choice i.e. m=30, y = 20

Note:-The number in the numerator must be less than the denominator if it is not less than the denominator

you will have error i.e. ^y/_m = ²⁰/₃₀

[ON] [2] [0] [SHIFT] [STO] [M] (M=20)

[AC] [3] [0] [SHIFT] [STO] [Y] (Y=30)

sin x = ^y/_m

[AC] [SHFT] [sin^-1] [ALPHA] [Y] [a^b/_c] [ALPHA] [M] [SHIFT] [STO] [X] (X=41.8103149)

Cos x = ? 

[AC] [cos] [ALPHA] [X] [=] (0.745355992)

Checking option A, B, C, D if any of the options gives 0.745355992 it is correct but if it did not give 0.745355992 it is wrong

Checking option (A) y² – m²

                                      m

[AC] [(] [ALPHA] [Y] [x²] [-] [ALPHA] [M] [x²] [)] [a^b/_c] [ALPHA] [M] [=] (-16.66666667)

Checking option (B) (A) √y² – m²

                                               m 

[AC]  [(] [√] [(] [ALPHA] [Y] [x²] [-] [ALPHA] [M] [x²] [)] [)] [a^b/_c] [ALPHA] [M] [=] (math error)

Option “B” is wrong

Checking option (C)  √m² – y² 

                                      m

[AC] [(] [√] [(] [ALPHA] [M] [x²] [-] [ALPHA] [Y] [x²] [)] [)] [a^b/_c] [ALPHA] [M] [=] (0.745355992)

Option “C” correct                                                      

(5). Evaluate sin² 0 + cos² 0 

                              cos² 0 

(A)¹/_cos² ₀  (B)sin² 0  (C) tan² x/_cos²₀   (D) cosec² 0

Solution:-Using CASIO fx-991MS calculator

Since the CASIO calculator doesn’t have “0” make 0 any of the alphabet in the CASIO fx-991MS calculator since the calculator has alphabets (A, B, C, D, E, F, X, Y, M) I.e. 0 = x also make x = any number of your choice i.e. x =30

[ON] [3] [0] [SHIFT] [STO] [X] (X=30)

[AC] [(] [(] [sin] [ALPHA] [X] [)] [x²] [+] [(] [cos] [ALPHA] [X] [)] [x²] [)] [a^b/_c] [cos] [ALPHA] [X] [x²] [=] (1.333333333) press [a^b/_c] (1⅓)

Checking option A, B, C, D if it gives you 1⅓ the option is correct if it didn’t give 1⅓ it is wrong

Checking (A)¹/_cos² ₀

since 0 = x :. ¹/_{(cos x)}² 

[AC] [1] [a^b/_c] [(] [cos] [ALPHA] [x] [)] [x²] [=] (1⅓)

The correct option is “A”                                                                               

(6). Simplify   sin 45 cos 30 – cos 45 sin 30

(A) sin 75 (B) sin 30 (C) sin 15 (D) cos 30

Solution:-Using CASIO fx-991MS calculator

[ON] [sin] [4] [5] [cos] [3] [0] [-] [cos] [4] [5] [sin] [3] [0] [=] (0.258819045)

Checking  option A, B, C,D if the option give 0.258819045 the option is correct but if it did not give 0.258819045 it is wrong

Checking option (A) sin 75

[AC] [sin] [7] [5] [=] (0.965925826) ;

Option (A) is wrong

Checking option (B) sin 30

[AC] [sin] [3] [0] [=] (0.5) ;. 

Option (B) is wrong

Checking option (C) sin 15

[AC] [sin] [1] [5] [=] (0.258819045) ;. 

Option (C) is correct

Checking option (D) cos 30

[AC] [cos] [3] [0] [=] (0.866025403) ;

Option (D) is wrong

(7). If 6cos² X + 2sin² X = 5 Find tan² X

(A) ⅓ (B) ¾ (C) ¼ (D) ⅔

Solution:-Using CASIO fx-991MS calculator

But sin² X = (sin X)²,  cos² X = (cos X)²,  tan² X = (tan X)²

Left axis is = right axis. Left axis=6cos²X + 2sin²X ,,Right axis = 5

To trace the answer make tan² X = any of the option given (A,B,C,D) Given if the left axis is equal to the right axis the option is correct but if the left axis is not equal to the right axis the option is wrong

checking option (A) ⅓   make tan² X = ⅓ (tan X)² = ⅓ tan X = √(⅓)

--Note when square rooting  a fraction put brackets to avoid errors using CASIO fx-991MS calculator                                                                                                

[ON] [SHIFT] [tan^-1] [√] [(] [1] [a^b/_c] [3] [)] [SHIFT] [STO] [X] (X=30^o)

Note:-Please use the key [AC] during calculations after you press the key [ON] on any question given                                                                                                                                                                  

6cos² X + 2sin² X  => 6(cos X)² + 2(sin X)² = >

[AC] [6] [(] [cos] [ALPHA] [X] [)] [x²] [+] [2] [sin] [ALPHA] [X] [x²] [=] (5)

Option (A) is correct

Checking option (B) ¾      make tan²x = ¾  (tan X)² = ¾ tan X=√(¾)

[AC] [SHIFT] [tan^-1] [√] [(] [3] [a^b/_c] [4] [)] [SHIFT] [STO] [X] (X=40.89339465^o)

Press [↑] [=] (4.285714286)

Option (B) is wrong

Checking option (C) ¼       make tan² X = ¼  (tan X)² = ¼  tan X=√(¼)

[AC] [SHIFT] [tan^-1] [√] [(] [1] [a^b/_c] [4] [)] [SHIFT] [STO] [X] (X=26.56505118^o)

Press [↑] [=] (5.2)   

Option (C) is wrong

Checking option (D) ⅔    make tan² X = ⅔  (tan X)² = ⅔  tan X=√(⅔)

[AC] [SHIFT] [tan^‑1] [√] [(] [2] [a^b/_c] [3] [)] [SHIFT] [STO] [X] (X=39.23152048^o)

Press [↑] [=] (4.4)

Option (D) is wrong

Since left axis = right axis . The correct option is option (A)

(8). Given that tan (x + y) = ¾ and tan x = ½ evaluate tan y

(A) ⅔ (B) ²⁄₁₁ (C) ¼ (D) ¾

Solution:-Using CASIO fx-991MS calculator

tan (x + y) = ¾ , left axis = tan (x + y), right axis = ¾ if the left axis is equal to the right axis the option is correct but if it is not equal to right axis the option is wrong   To trace the answer in the option given make tan x = ½ and tan = any of the option given since we should find tan y :. Tan x = ½ 

[ON] [SHIFT] [tan^-1] [1] [a^b/_c] [2] [SHIFT] [STO] [X] (x=26.56505118)

Checking option (A) ⅔   : Let tan y = ⅔

[AC] [SHIFT] [tan^-1] [2] [a^b/_c] [3] [SHIFT] [STO] [Y] (y=33.69006753)

Tan (x + y) = ?

[AC] [tan] [(] [ALPHA] [X] [+] [ALPHA] [Y] [=] (1.75) :.

Option “A” is wrong

Checking option (B) ²⁄₁₁  let tan y = ²⁄₁₁

[AC] [SHIFT] [tan^-1] [2] [a^b/_c] [1] [1] [SHIFT] [STO] [Y] (y=10.30484647)

Press [↑] [=] (0.75) press [a^b⁄_c] [=] (¾)

Option “B” is correct You can use this format shown to solve option “C’ and “D” but option “C” and “D” is wrong the correct option is “B”

Unforgiveness is a sin, if you don’t forgive; your trespasses (sins) will not be forgiven

Reason. In Mathew 6:14-15- if ye forgive men their trespasses your heavenly father will also forgive your trespasses but if ye forgive not men their trespasses neither will your father forgive your trespasses

Also in Mark 11:25-26- And when ye stand praying forgive if you have ought against any that your father also which is in heaven will forgive your trespasses. But if ye do not forgive, neither will your father which is in heaven forgive your trespasses.

For your heavenly father to forgive your trespasses. You must forgive those that trespass against you including your enemies. In Matthew 5:44-Love your enemies, bless them that curse you, do good to them that hate you, pray for them which despitefully use you and persecute you. As a Christian/Believer of JESUS CHRIST you must forgive your enemies including those that commit evil against you because if you don’t forgive them you are in darkness. In 1 John 2:9-11 – He that saith he is in the light and hateth his brother is in darkness even until now. He that loveth his brother abideth in the light and there is none occasion of stumbling in him. But he that hateth his brother is in darkness and walketh in darkness and knoweth not wither because that darkness has blinded his eyes. A believer will say I am in the light by being a deacon, choir, usher, pastor, evangelist, But know that Christians/believers who is indulge in unforgiveness will not inherit the kingdom of God and they will spend their eternity in hell (the fire that never shall be quenched). Mark 9:43

Know that if you don’t forgive today, your prayers will not be answered. But if you are touched by the word of GOD on unforgiveness and you have make a decision in your heart to forgive whomsoever wronged you, offend you, please make this confession

LORD JESUS, as I come to you, have mercy upon me, as I forgive those that offend me today. Forgive me my trespasses; receive me into your kingdom, IN JESUS NAME AMEN.

NOW: that you have confess this prayer of unforgiveness you are free to confess the next prayer below if you are not

Born-again

if you have not born-again or given your life to Jesus Christ say this prayer “Lord JESUS, as I come to you today, Have mercy upon me, Forgive me all my sins, wash me with your precious blood, save my soul, as I receive you today as my lord and personal savior, write my name in the book of life, IN JESUS NAME AMEN.

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